Can I paint a door without removing it from the hinges?
Sure, but you increase the odds of drips and runs that way. 

Either way you have to leave it open until the paint dries. 

If you take it off, you can bring it to a warm part of the house so it dries faster. Might not be much different on a warm/hot day.

Be sure to put a dropcloth under it to keep paint off the floor.