I bought a new Hampton Bay ceiling fan that came with three 14-watt CFLs. My fan and lights are on a remote and the CFLs flicker when they are turned "off" with the remote. I want to replace the CFLs with regular incandescent bulbs. Since the fixture says max 14W CFL per light, can I safely use 60 watt incandescent bulbs instead?

I presume you can.

Did the fan come with the remote or did you buy it separately?

Ceiling fan remote controls - help and FAQ- Ceiling Fans N More

I bought the remote separately.

If you want to use the remote, you will have to use incandescents. Personally I would pick the CFLs over the remote, as you will have problems with the remote down the road.