No, except for when they do
"Lingering effects" is a pretty broad topic, including at least Conditions, spell effects, disease, madness, curses, ability score reduction, maximum hit point reduction, and 'burning or whatever'. Many of these already have answers on this site (which I have linked). Some do not have a clear consensus.
Conditions
By far the best documentation within RAW comes from Conditions. I'm using a big-C on Conditions to specify it as a game term that includes the 15 Conditions listed in Appendix A of the PHB, and not as a natural English 'conditions' which might include other effects. Appendix A tells us that:
A condition lasts either until it is countered (the prone condition is countered by standing up, for example) or for a duration specified by the effect that imposed the condition.
Since the rules for Wild Shape, and the Conditions themselves, don't specify that the new shape "counters" any Conditions, in general we can assume that Conditions carry over to the new form.
One exception might be when the new shape is immune to the condition. If you wild shape into an elemental, you will at the very least not be affected by having the poisoned condition because your new form is immune to poison. RAW, it is not clear whether the Condition to which your form has immunity is lost, or merely suppressed. If suppressed, you might have the Condition, but not be affected by it - or you might not have the Condition in your wild shape, but it would return if you ended your wild shape during its duration. Unfortunately, it is not clear RAW what happens to a Condition if you become immune to the Condition while suffering from it.
In their answer to this question, Dale M cites a Crawford tweet as evidence that if the wild shape form is immune to the Condition, then the Condition ends:
In #DnD, the exceptional trumps the general. (No longer being a valid target trumps condition carryover.)
However, it is not clear what Crawford means here by "trumps". It could just mean that the condition is suppressed rather than removed, as he elsewhere suggests for spell effects:
There's no rule governing what happens when a valid spell target temporarily becomes an invalid target. A good rule of thumb is that the spell is suppressed while the target is invalid.
Does Exhaustion carry over from a druid's wild shaped form to their regular form? (asked here) Yes, exhaustion (and being poisoned) are both Conditions, and as such are carried over (unless the new form is immune, in which case it is not clear, as discussed above).
A further complication is that sometimes Conditions are caused by spell effects, and it is not clear whether the Conditions themselves persist when the spell effects are suppressed. Suppose, for example, your druid was under the influence of a Charm Person spell, which has as one of its effects imposing the charmed Condition "until the spell ends [duration 1 hour] or until you or your companions do anything harmful to it." Another effect of the spell but not related to the Condition is that your druid regards the caster as "a friendly acquaintance". If your charmed druid then wild shapes, their new beast shape would not be a valid target for the spell (which targets only humanoids), but would have no immunity to the preexisting charmed Condition itself. If we follow Crawford's 'rule of thumb' and say that the spell is suppressed rather then ended, then the 'friendly acquaintance' spell effect is suppressed as well. But does the charmed Condition remain, so that you are still unable to attack the caster? Conditions, we know, last until they are "countered" - but does suppressing the spell that generated the charmed Condition 'counter' the Condition? RAW doesn't say.
Spell effects
After Conditions, the effect of wild shape on spell effects is the best documented in RAW and RAI (through multiple, specific tweets) of all potential effects. We know that, like Conditions, spell effects are assumed to remain on their target unless something fulfils one of their end conditions (and for spells which require Concentration, transforming does not break it). Thus, for example, a druid who cast barkskin while in their natural form would still have it apply when they assumed their wild shape. Likewise for a pre-cast mage armor, or a guardian of nature spell, a shield of faith, and mostly any other spell. Some exceptions include spells that specifically affect the druid's original form once only but don't perpetuate after the change - which might include enlarge/reduce. Another exception would be spells for which the druid's new form would not be a valid target. Here the RAW are silent, but Crawford's recommendation, as cited above, is that the spell would be suppressed (and you can read more about that here and here).
Disease
Being diseased is not a Condition in 5e. Each disease has its own particular rules and mechanics rather than being bound by an overall approach (DMG 256):
the specifics of how a disease works aren't bound by a common set of rules.
This means that we cannot assume that disease generally works like a Condition and is assumed to carry-over unless countered. Further, 5e plays with our real-world concept of infectiousness by explicitly stating that diseases might or might not pass between individuals of different types (ibid):
Diseases can affect any creature, and a given illness might or might not pass from one race or kind of creature to another. A plague might affect only constructs or undead, or sweep through a halfling neighborhood but leave other races
untouched. What matters is the story you want to tell.
Of the three example diseases given in the DMG (257), Cackle Fever affects only non-gnome humanoids, Sewer Plague affects only humanoids but can be carried by beasts, and Sight Rot affects both humanoids and beasts. Thus your druid might be able to wild-shape his way out of the first two but not the third. The Conditions imposed by the diseases (incapacitation and exhaustion), however, would carry over into the new form if they were already present (and it is likely that the druid couldn't wild shape while incapacitated anyway). Although many things remove diseases in general, transforming between different physical forms is not one of them. For diseases other than the three listed in the DMG, it would be entirely up to the DM and 'the story they want to tell' whether wild shaping would remove the disease or not (unless the rules description of the specific disease said that it could affect beasts). If the disease was contracted while in wild shape, it would also be the DM's decision whether returning to the druid's natural form would remove it.
Madness
Madness is neither a Condition, spell effect, nor a disease (although some diseases and spell effects can cause madness) but rather is its own thing. Very few things in the game are capable of "Curing Madness" (DMG 260), and changing physical forms is not listed among them, so Madness would persist in the new wild shape.
Curses
As a spell effect, if your druid is the target of a Bestow Curse, the effect of the curse would persist despite the wild shape (here, although the question is asked for polymorph, not wild shape).
You are also unlikely to be able to use wild shape to rid yourself of a cursed item, since generally you will either be unwilling to part with the item due to the curse, or if you can part with it the curse will persist anyway. Since your wild shape says that
Equipment that merges with the form has no effect until you leave the form
then at least the effect of the cursed item would not apply while you were in wild shape. As one interesting irony, merging with the cursed item might suppress the effect that makes you unwilling to part with it, but you would be unable to part with it while it was merged. Returning to your natural form would allow you to be able to drop the item - but you would then be unwilling to do so. Perhaps your wild shape has some way to communicate to other party members that they should ready a remove curse for when you change back.
Ability Score Reduction
There aren't a lot of effects that are going to reduce ability scores, and since the druid retains three of their own scores in wild shape we need only worry about whether wild shape carries over changes to Str, Dex, and Con. Strength is probably the most frequent ability score reduced, due to the effect of the Strength Drain from Shadows:
Strength Drain. Hit: 9 (2d6 + 2) necrotic damage, and the target's Strength score is reduced by 1d4...the reduction lasts until the target finishes a short or long rest.
Thus we can ask whether a strength-drained druid would have reduced strength in wild form. Different abilities may use different language, but notice that here neither the wording of wild shape nor Strength Drain suggest that wild shape would prevent the ability score change from carrying over so long as the druid may reasonably be considered still 'the target' of the original drain when in wild shape. Thus the wild shape would have a new base strength in accord with its new form, but reduced by the same amount the druid's natural form Strength had been been.
Hit point maximum reduction
This is going to fall to DM interpretation, as we have no clear guidance. Using the succubus' ability as an example:
Draining Kiss. The fiend kisses a creature charmed by it or a willing creature. The target must make a DC 15 Constitution saving throw against this magic, taking 32 (5d10 + 5) psychic damage on a failed save, or half as much damage on a successful one. The target's hit point maximum is reduced by an amount equal to the damage taken. This reduction lasts until the target finishes a long rest. The target dies if this effect reduces its hit point maximum to 0.
Note that this ability refers to both "a creature" and "the target", and specifies an end condition to the hp maximum reduction effect: "This reduction lasts until the target finishes a long rest." By analogy with the rule that Conditions specify their own end, RAW the hp max reduction should remain regardless of what form the druid is in, so long as this form can be considered the same creature and the same target, as it certainly is for every other lingering effect we have considered so far.
Meanwhile, the description of wild shape says:
Your game statistics are replaced by the statistics of the beast, but you retain your alignment, personality, and Intelligence, Wisdom, and Charisma scores...When you transform, you assume the beast’s hit points and Hit Dice. When you revert to your normal form, you return to the number of hit points you had before you transformed.
Thus we know that your hp change when you change forms. The passage quoted does not say that your hp maximum changes, but we know how to calculate the hp maximum for a character, and we know that the hp maximum for a wild shape is listed in its stat block, and that since these two are derived differently the hp maximum must change as well. Further, changing form implies a change in the constitution score (if not the specific number, at least how it is determined), and that
If your Constitution modifier changes, your hit point maximum changes as well, as though you had the new modifier from 1st level...Or if you're 7th level and some effect lowers your Constitution score so as to reduce your Constitution modifier by 1, your hit point maximum is reduced by 7.
So, while your hp maximum itself clearly changes with your form, nothing RAW suggests that effects that reduce your hp maximum do not persist when you change form, like any other lingering effect. If your old body was prone when you changed, your new body would still be prone, even though it is a different body, because nothing says that wild shape changes the prone Condition. If your old hp maximum had an applied reduction, then your new, different hp maximum would still have the reduction, because nothing in the wild shape rules says that it would be removed or suppressed. That is the reasoning behind the accepted answer to this question.
However, in direct contrast to this reasoning, we have a direct and specific tweet from Crawford:
Wild Shape—a reduction to hp maximum doesn't carry over from your beast form to your true form or vice versa.
This seems, at least to me, to be a de novo interpretation by Crawford and not tied to the existing rules. In fact, when pressed as to why the reduction doesn't carry over, he offers a string of (in my opinion) self-contradictory justifications that don't make reference to existing rules. Nonetheless, the presence of such a clear and direct RAI statement appears to be sufficient to have this answer, stating that a reduction in hp maximum does not carry over, accepted for this question.
Thus, even within rpg stacks we have no consensus on this matter, so a DM will need to decide which interpretation to implement.
Burning or whatever
Let's consider alchemist's fire:
On a hit, the target takes 1d4 fire damage at the start of each of its turns. A creature can end this damage by using its action to make a DC 10 Dexterity check to extinguish the flames.
This refers to both "the target" and "a creature". Again, so long as the new wild shape form is considered to be the same target and the same creature, as it is in every other lingering effect, there is no reason that wild-shaping would turn an on-fire druid into a not-on-fire beast. Unlike an intact bottle of alchemist's fire being carried by your druid, the flames of alchemist's fire which had hit them would certainly not be considered "your equipment" and thus eligible for you to either drop or merge into your new form.
No, except for when they do
To summarize lingering effects:
Conditions - Carry over unless the new form is immune (and then not clear whether they are lost or suppressed)
Spell effects - Carry over unless the new form is immune (and then not clear whether they are lost or suppressed) or is no longer a legitimate target (and then not clear whether they continue or are lost or suppressed)
Disease - DM's decision as to whether the new form can suffer from the disease
Madness - Carries over
Curses - Cursed items carry over, for the spell see spell effects
Ability score reduction - Carries over
Hp maximum reduction - DM's interpretation
Burning - carries over
