What's the fastest way to strip paint off of a piece of installed trim/moulding?
The least time route would definitely be the "remove the trim and replace with fresh" route.

However, this would depend on you being able to get a suitable replacement.

Heat gun is quite quick, but you have to be careful of a) the chemicals released when heating the paint and b) not scorching the wood. Heat guns shouldn't really be used where you've got lead based paint.

Chemicals will be useful where you've got delicate/intricate mouldings but will probably take the longest as you might need several applications if you've got lots of layers of paint.

You will definitely need to sand the mouldings if you use a heat gun, you may have to if you use chemicals. This might not be practicable for delicate/intricate mouldings.