I have to install a prehung door where the floor is finished and I have to cut down the door as the rough opening is less than the standard height. I don't want to cut much off the door, so how much room do I need to leave at the top?

What kind of flooring is it? I assume there is no threshold at the doorway. Regardless, you want to make your cut at the bottom of the door (and the frame if needed).

Is it a prehung hollow core interior door? Look at the other doors in the house and you'll get an idea of the clearance necessary for the new door. If it's a mobile or manufactured home the space under the door can be important when it comes to circulating air and HVAC balance.

Good luck and remember to put some tape (masking or duct) on the door along your intended cut-line (I do it on both sides of the door but it's the bottom of the cut where the blade will chip the door skin).

The standard distance between the top of the door and the frame is 1/8". If you are installing a pre-hung door you won't be able to cut the top of the door. If you did you would have to route new hinge mortices and patch the old ones. If you are going to do all that, you may as well buy a door and build your own frame. Split jam frames are available for standard wall thicknesses.
Door frames are not normally nailed to the header inside the wall. There can be as much as 2" between the top of the frame and the header. Remove the existing frame before deciding what the maximum height is.

If you are installing a standard, pre-hung, hollow core door, the bottom rail is about 1 to 1 1/2" wide. If you have to cut off more than that you will have to replace the bottom rail. The bottom rail of a standard 1 3/8" hollow core door will be 1 1/8" thick. Machine a new rail from straight grain pine or fir, 1 to 1 1/2" wide and 1 1/8" thick. Cut to length so that you have a tight fit between styles. Install using glue. Coat the inside of both sides of the skin with glue, coat both sides of the rail with glue. Install inside the door between the skins. Clamp with lots of clamps. You can nail the styles to each end of the rail but it is really not necessary. Most important thing is to bring the skins into contact with the rail for good adhesion. You can use a lot of spring clamps or you put a piece of 1 by material on each side of the door and clamping at the ends and the center.
For the neatest job, cut the door about 1/4" long, install the new rail and then after the glue has dried, cut off the door to the correct height.

HK petty much nailed it.

If u are asking how much room should be left between the rough opening header and the door jamb header. This can be as little as 1/2''. If the jamb legs have ears above jamb header, they can be removed for a tighter fit.

Also the old bottom rail can be cleaned and reused.Peel off door veneer and scrap old dry glue with a multi scraper. Reinstall as HK described.

Chuck