Let's say I cast hold person on a foe. While he is paralyzed, can I grab something he is holding in his hand?
If so, what would the check be?
Does he even get to resist it?
Yes, you can try this
On your turn, you can take one of the standard actions, an action enabled by a class feature or other ability, or improvise an action. Trying to disarm/steal from someone who's paralyzed is a reasonable improvised action.
Improvised actions, of course, don't have much in the way of explicit rules regarding their function. The DM might make up rules for disarming characters, and then apply advantage because the foe can't resist, or the DM might just let you do it, or require a STR check or a DEX check or something else altogether. What happens here is undefined.
Paralyzed (Appendix A, PHB, p. 291)
- A paralyzed creature is incapacitated (see the condition) and can’t move or speak.
- The creature automatically fails Strength and Dexterity saving throws.
- Attack rolls against the creature have advantage.
- Any attack that hits the creature is a critical hit if the attacker is within 5 feet of the creature.
Incapacitated (Appendix A, PHB, p. 290)
- An incapacitated creature can’t take actions or reactions.
All of the citations below are from Using Ability Scores in Chapter 7 of the PHB
The GM calls for an ability check when a character or monster attempts an action (other than an attack) that has a chance of failure. When the outcome is uncertain, the dice determine the results. (p. 174)
Your GM may rule that there is no chance of failure, and you simply grab it.
A reason to rule this way is that the paralyzed foe cannot taken actions, nor reactions, and can't move.
A reason not to rule this was would be that the foe's grip on this item is still being applied, or was never released. (The conditions paralyzed and incapacitated do not state that all items held are dropped, for example).
If the ruling is that you don't simply grab it using an action (or interaction with an object), the GM will likely adjudicate it as an ability check (strength or dexterity) or a contest.
Sometimes one character’s or monster’s efforts are directly opposed to another’s. This can occur when both of them are trying to do the same thing and only one can succeed, such as attempting to snatch up a magic ring that has fallen on the floor. This situation also applies when one of them is trying to prevent the other one from accomplishing a goal — for example, when a monster tries to force open a door that an adventurer is holding closed. In situations like these, the outcome is determined by a special form of ability check, called a contest. (p. 174)
Whether it's a contested Strength check or a contested Dexterity check will depend in the circumstances of this particular attempt and the GM's assessment of how you are attempting to grab the item.
The GM can also decide that circumstances influence a roll in one direction or the other and grant advantage or impose disadvantage as a result. (p. 173)
The circumstance of the foe being incapacitated argues for your attempt having advantage, but nothing explicitly requires the application of advantage since this attempt isn't an attack (see the conditions above). However, because you get advantage on attacks versus this paralyzed foe, granting you advantage on an opposed ability check is a reasonable ruling.
Depending upon circumstances, either you spend an action and you get to grab it (most likely ruling) or I grant you advantage on a Dexterity or Strength check (more complicated situation/battle), and in an unusual case an opposed Strength or Dexterity check. Your DM may rule similarly, or otherwise. @Damon makes a good point on the imprecision of what "paralyzed" means in D&D 5e versus real life paralylsis, and previous edition descriptions of being held.
I've come across this in one of my games before as a DM. I ended up ruling that the paralyzed creature had a "Passive Strength DC" in much the same fashion as the rules for Passive Perception. 10 + Strength Mod, +5/-5 for Advantage and Disadvantage respectively.
Note: When I did this at my table, I did not rule that being paralyzed would give disadvantage to that check, and rather represents the very moment before the effect that caused the paralyzation. So a raging barbarian would carry his advantage into that calculation, or any spell that gave adv/dis would carry into the DC calc even after the effect expired, as long as the paralyzation was still active.
This gives somewhat sensible results (creatures that would have a reason to be hanging on tighter when they're paralyzed have higher DCs) and gives more opportunities for my players to cooperate on certain tactics.
I would say "Yes, but...".
Paralysis in D&D terms -- in every version that I'm aware of -- is not equivalent to what the common meaning of the word would suggest (something in between tetraplegia and locked-in syndrome). Limbs are not atonic, paralyzed creatures/people do not drop everything and fall prone as they would in reality.
Unluckily, D&D 5e is not nearly as specific in its description as the previous versions such as e.g. 3.5e were. There is no mention of "frozen in place" as in the Hold Person and Paralysis description, or "rooted to the spot, frozen and helpless. Not even friends can move his limbs" as in the supernatural ability description.
All it really says is that the creature is incapacitated and automatically fails all strength checks, attacks have advantage, etc.
Thus, from a RAW point of view, I guess you can certainly just take anything the paralyzed victim is carrying, including something the victim is holding.
However, from a more "logical" point of view, even though this is not explicitly said, the creature is most likely rigidly paralyzed (spasmic), so removing something held in the creature's hand will require breaking the victim's fingers...