I need to trim off 1/2 to 3/4 inch from the bottom of an interior hollow door to negotiate with the new tile floor, but I'm concerned about compromising the integrity of the door once I do that. Is there some sort of cap I can place on the bottom of the door to fill the gap once I trim the bottom off? What should I do here? Thanks.

There is usually a solid piece of wood at the bottom of hollow doors so that the door can be trimmed as you describe. You should be OK with trimming off 1/2" - 3/4".

A bigger problem is using the right blade, such that splintering of the venier doesn't happen.

It would probably require a fine tooth blade and cut from both sides.

clamp a straight edge on the bottom of the door to guide your circular saw so you get a straight edge. (straight edge = any piece of reasonably straight lumber)

you can also score the top side along the cut line to reduce the splinters. The bottom edge will be fine because of the way the blade turns. (i.e. put the good side of the door, if there is one, face down). If your blade is sharp, it shouldn't be a problem.

As Amricca says, you should be OK to take out 1/2 to 3/4". If you end up having to cut the entire piece of wood at the bottom out, then strip and sand the old veneer from the wood, and reglue it inside the bottom again.

Well look at you, 3 great answers that are all correct and helpful. SDtart with rtw, get the blade KISS recommends and take the cut amricca says is safe, it is.