If I purchase a warrant or hit, or obtain a hit as a bonus from a conspiracy tile, do I HAVE to play it immediately? Can I hold on to it for a later time when I can use it more strategically?

Yeah, the wording on the locations is pretty unequivocal.

Allowing players to hang on to them might be an interesting house rule for warrants, which are fairly weak. Hits are too, given their cost and rarity, but if players could use them whenever they wanted, it'd be likely to severely nerf determining the murderer, since each game would probably end in a Day 12 dogpile on whichever suspect appears guiltiest.