Most TOS episodes are ambiguous on that, but "Metamorphosis" contains the single explicit bit: our heroes in the shuttlecraft observe the Companion cloud first heading towards them at warp speed, then "staying right with" them. This would make very little sense if the shuttle weren't at warp as well; certainly our heroes wouldn't express any sort of "it's staying with us!" amazement or concern if the Companion just performed the approach at warp, then slowed down to the less-than-warp speed they were maintaining.
I'd think anything with nacelles on it should be warp-capable. But the tiny TNG shuttlepods for some reason do not have that capability - or at least the specific craft used by the double Picard in "Time Squared" was said to lack it. (Perhaps it was broken? Or perhaps Riker just meant the shuttle did not have enough warp capacity to do the time travel trick they were discussing?)
As for performance, VOY gives the unseen Type 9 (perhaps the same as the big TNG Tech Manual cargo shuttle?) a speed of warp four. Runabouts seem to max out at warp five. So these much humbler craft should probably be limited to warp two or thereabouts. Mind you, that still appears to be a respectable interstellar speed in TOS and ENT.