Allowing sneak attack on improvised weapons could be unbalanced

In this situation there are four very distinct groups of improvised weapon that have different answers:

1. Improvised weapons that resemble finesse weapons (glass bottles, cutlery, fire pokers, etc.)


2. Improvised weapons that resemble other weapons


3. Improvised weapons that don't resemble any weapons


4. Items that require an improvised weapon attack (Holy Water, Vial of Acid, etc.).

An improvised weapon includes any object you can wield in one of two hands, such as broken glass, a table leg, a frying pan, a wagon wheel, or a dead goblin.

The improvised weapon rules gives us a series of example of what counts as an improvised weapon. The damage type for these improvised weapons will be the same as the real weapon they are most similar to. If they are similar enough they can actually be treated as a real weapon of that kind, as given by the following rule.

Often, an improvised weapon is similar to an actual weapon and can be treated as such. For example, a table leg is akin to a club. At the DM's option, character proficient with a weapon can use a similar object as if it were that weapon and use his or her proficiency bonus.

Improvised weapons that resemble finesse weapons

Any improvised weapon that is sufficiently similar to a finesse or ranged weapon can be treated as one. Therefore not only is it balanced to allow sneak attack on these weapons it is officially supported by the rules. These weapons will all deal a physical damage type; piercing, slashing or (rarely) bludgeoning which is a normal limitation of sneak attack.

Improvised weapons that resemble non-finesse weapons

For improvised which resembled larger, non-finesse, weapons allowing sneak attack may be slightly unbalanced. For instance allowing sneak attack on attacks with a tree branch treated as a great-club would allow a multiclassed character to use both sneak attack and great weapon fighting on the damage roll of the attack. Re-rolling 1's and 2's on sneak attack would be extremely strong.

Improvised weapons that don't resemble any weapon

An object that bears no resemblance to a weapon deals 1d4 damage (the DM assigns a damage type appropriate to the object).

In damage terms these truly improvised weapons are the same as daggers other than the fact they lack the finesse property. By RAW you would not be able to sneak attack them, however in damage balance terms there is no issue allowing it. Even with feats to improve improvised weapons these will always be inferior to a simple dagger.

Items that require a improvised weapon attack

From the description of Holy Water we get:

Make a ranged attack against a target creature, treating the holy water as an improvised weapon.

This rule describes how to calculate the attack roll for holy water. Specifically it says to treat the attack as an improvised attack, not treat the holy water as an improvised weapon which resemble a ranged weapon you are proficient with.

Allowing sneak attack on items like this opens the possibility of sneak attack dealing non-physical damage types. As the physical damages types are commonly resisted at higher CRs this would be a bump to the power of the sneak attack feature. Allowing rogues to deal large amounts of radiant damage significantly steps on the toes of clerics and paladins (as per Someone_Evil's answer) and other items (i.e. Alchemists Fire) would allow them to step on the toes of other casters.

As David Coffron points out allowing sneak attack on items that only require an object interaction and not at Attack action is very strong. The thief's fast hands feature would allow sneak attack on a bonus action in addition to on a readied action. Multiple sneak attacks per round is a huge boost to damage output and allowing it so easily would be unbalanced.

Thematically it's wrong

A side note to the balance concerns, thematically sneak attack is 'precision damage'. There is no way to be precise with a splash weapon. I don't even the DM who has to figure out how to narrate sneak attack from a character throwing a table at something.

This wouldn't affect your party, but I'm answering as it could help someone else with a similar query.

This makes the Fast Hands feature very powerful.

The Thief Roguish Archetype has the feature Fast Hands which says:

you can use the bonus action granted by your Cunning Action to ... take the Use an Object action.

Two objects of particular note are acid and alchemist's fire. A thief rogue who can throw two of these in one turn is particularly effective since these items deal markably more damage then a simple weapon (or cancel out an enemy's action to avoid more damage in the case of alchemist's fire). The standard downside to this tactic is you pass up the opportunity to deal Sneak Attack damage.

Usually, a thief rogue employing this tactic will make a weapon attack and then throw the item with their bonus action. This only allows for one chance to hit for Sneak Attack, while you could throw with both your action and your bonus action without sacrificing the chances to Sneak Attack using your proposed rule.

Additionally, the rogue could use a bonus action to throw the item, and then take the Ready action to throw another (or make an attack) with some trigger that would occur on someone else's turn. This would allow for using Sneak Attack twice in one round, while that is usually impossible without some other method of attacking with your reaction (thanks @Chris Starnes in the comments).

Depends on the opposition they're meeting.

The main thing you are allowing is letting the Rogue deal its damage as radiant (at whatever cost the holy water has to them) which isn't something they normally get to do. So if they are encountering a lot of things with vulnerability to radiant or resistance to non-magical attacks (as is fairly common among fiends and undead), the rogue will do more than they normally would against those. If they aren't encountering such opponents, throwing holy water probably doesn't come up at all.

Another concern is whether allowing rogues to deal significant radiant damage steps on the toes of Paladins or Clerics, who would normally be the ones to deal radiant damage (See Paladin's Divine Smite feature and the guiding bolt spell as examples) and so this house-rule would let rogues steal (hah!) the moments where Paladins and Clerics get to be extra powerful and cool. (This is possibly something you will want to talk to your players, particularly the Paladin, about.)

I don't think it would unbalance the game, because it's A) fairly costly for the damage it does, B) has quite limited range, and C) already requires an action to throw just one, so can't benefit from multiple attacks or "being used as an item".

Honestly it seems like a cool idea, and it rewards player creativity. Unless they're all level 1 and they have a hundred vials of holy water, 2d6 damage isn't a crazy amount, especially given the other limitations. (Also, I assume this is being used by the undead mastermind rogue? Carrying a bunch of fragile vials of holy water, as an undead? That has the potential to go VERY wrong if they trip...)

Edit, due to complaints about the answer only dealing with Holy Water:

Improvised Weapon, according to RAW, leaves it up to the DM to set the damage die and type for the object in question, as well as determining if it is close enough to an actual weapon to be counted as such. If not, they don't get their Proficiency Bonus to the attack roll. So, normal weapons will in most cases still be better. I find it hard to imagine scenarios where it could be badly abused, if the DM is making reasonable calls on what can be used. (Also, it's fun and thematic!)

Regarding fears that the Rogue Fast Hands ability will significantly unbalance the game, I don't see that being the case. You can only do Sneak Attack damage once per round anyway. Further, it's dependent on having Advantage on the Attack Roll, which further limits their options.

Conversely, if you want an excuse to disallow it, it could be argued that Holy Water is magical, and thus subject to the rule on DMG p.141:

If an item requires an action to activate, that action isn't a
function of the Use an Object action, so a feature such as the rogue's
Fast Hands can't be used to activate the item.

Further, I would also argue that throwing a splash attack into someone's face could very reasonably be considered a precision attack.