My ceiling has red, black, white, and copper wiring. My remote has only red and white. Do I cap the black? If not, what do I do with it?

It all depends.

It depends on how the wall switch is wired. Remove the switch cover. You will probably find two white wires connected together. You will probably find a black wire connected another black wire and connected to the switch. There will also probably be a red wire connected to the switch. If that is correct, you connect the black wire in the ceiling to the red fan wire and connect the two white wires together. You then cap the red wire with a wire with a wire nut.

If you find a black wire connected to a red wire and connected to the switch and another black connected to the switch, you cap the black.

In other words, cap the black wire if you have two black wires connected to the switch. Cap the red wire if you have a black and red wire connected to the switch.

OK you must be more clear. Your remote has red and white, on which side is that and what is the other side. A remote receiver has an incoming feed side and an output side. Is there a second wa;ll switch somewhere that may also control the fan or was there a light kit?