My long-term RPG instinct is to say “clearly they don’t”. (Pathfinder, 3.5, AD&D, etc.)
Looking at some NPCs seems to be back this up. The Veteran (MM 350) deals a straight 1d10 with their Heavy Crossbow, despite having Dex+1, and the Scout (MM 349) deals 1d8+Dex with their Longbow.
However, I cannot actually find where this is detailed in the PHB…
- Section on Dexterity (PHB 176) does not have any exception for the crossbow.
You add your Dexterity modifier to your attack roll and your damage roll when attacking with a ranged weapon, such as a sling or longbow.
- Crossbow features are Ammunition and Loading, but neither of these seem to address damage. (PHB 146-9])
- The sections on “Making an attack” (PHB 196-8) don’t mention anything about damage modifiers at all.
Also, I cannot find any mention in the latest errata (v1.1) (Oct 2015).
Does anyone else have a RAW for “crossbows don’t add Dex to damage?”.
UPDATE: the MM has a few other examples that are not consistent with “Dex on damage”, but in general, most are accurate. The answer below seems to back my findings, crossbows do add Dex modifier to damage.
UPDATE 2: the MM errata (2015-12) corrects the Veteran example.