I have a specific character concept in mind but for it to work I would need to change the UA Psionic Rogue's Psychic Daggers into Fire Daggers. I plan to ask my DM about it but before I do I want to be aware of any possible balance implications in the damage type change. Would it be over- or under-powered or would it be relatively the same?
The Balance Implications would probably be minimal
In my experience as a DM for several different settings, the range of creatures that typically have resistance or immunity to Fire Damage is not meaningfully more or less powerful than the (comparably few) creatures that typically have resistance or immunity to Psychic damage, nor are they especially more or less frequently encountered.
I do want to acknowledge this answer, which summarizes vulnerabilities, resistances, and immunities by damage type; a casual observation from which is that Fire Resistance/Immunity is far more common than Psychic. However, I also want to point out that that answer can be misleading, because it's stacked for some of those elements with very specific creature types that all tend to have similar resistances/immunities; and most of those creatures tend to be very infrequently encountered (Demons/Devils, Dragons, etc.) that aren't going to represent the majority of encounters you face.
Or, put more succinctly: that table is more accurate if your campaign is expected to have quite a lot of Demons and Dragons in it. Less so if, like most campaigns, it skews more towards humanoid encounters with a non-humanoid final boss.
Having said that...
This will probably make the feature marginally weaker, in General
Although it's not common for creatures to have damage resistances at all; and when they do, it's not common that it will skew towards Fire more or less than Psychic (in my experience), it is true that many features, especially spells, are more likely to grant resistance to Fire damage than they are Psychic Damage. Spells like Absorb Elements (XGE, 150) or Protection from Energy (PHB, 270) are limited to the 5 damage types of Acid, Cold, Fire, Lightning, or Thunder Damage, with Poison, Necrotic, Radiant, Psychic, and Force damage all not being targetable by those spells.
By this metric, it does become more likely that a feature which deals Fire Damage is more likely to be resisted than a feature that deals Psychic Damage (because fighting an enemy spellcaster, humanoid or otherwise, actually does happen relatively commonly). So in this context, changing the damage type to fire has a potential to weaken the feature.
But this is pretty marginal, and in general, I would not anticipate that this kind of change would majorly alter the balance of the character as a whole.
Multiclassing (probably) doesn't have an impact on the power of this feature
The inciting example is a spell like Elemental Bane:
[...] choose one of the following damage types: acid, cold, fire, lightning, or thunder. [...] The first time each turn the affected target takes damage of the chosen type, the target takes an extra 2d6 damage of that type. Moreover, the target loses any resistance to that damage type until the spell ends.
—Elemental Bane, Xanathar's Guide to Everything, pg. 155
In theory, this does represent a boost to the Rogue's damage output; but in practice this doesn't hold up.
Elemental bane is a 4th level transmutation spell. This means that for this Rogue to gain access to this spell, they are required to commit at least 7 levels into another [spellcasting] class in order to be able to cast it. In exchange for doing this, they've lost 3d6 or 4d6 from their Rogue sneak attack feature. This is a quite significant anti-synergy for a feature that, unless those levels are in Warlock, won't even be active in most fights.
And because both sneak attack and elemental bane limit themselves to triggering once per turn, it's not even possible to boost this effect through further multiclassing, like into Fighter to gain extra attacks.
So unless there's another spell or feature that is only obtainable through multiclassing and would interact with this damage enough to offset the lost damage from missing rogue levels, I don't think this amounts to any impact on the balance of this change.
Resistance and Immunity
Fire is the fourth (third if you exclude nonmagic weapon damage) most commonly resisted damage type as well as being second most common to be immune. Almost all fiends are resistant or immune, Fire elementals, Fire giants, and the list goes on. The only damage type less resisted than Psychic is Force (although Radiant has more resistances and fewer immunities)
You would be overall reducing your effectiveness against a large portion of D&D's most common villains (Fiends, Dragons, certain Giants).
I would say the balance difference depends on your campaign. If you're fighting a ton of Demons and Devils you'll lose out. If your campaign is undead focused you won't lose as much. Talk to your GM and ask them how many monsters with fire resistance they anticipate you fighting.
Do keep in mind that your sneak attack damage is based on the weapon damage, so rolling 12d6 fire damage sounds cool, it's not as good if it's halved.
The numbers I use come from This spreadsheet