Use the following trick to ride a motorcycle without Niko wearing a helmet. Tap the Triangle at any point during the animation where he puts the helmet on, but before the helmet actually appears. Doing this cancels the animation and Niko will not wear his helmet. If you wait too long and press that button after the helmet has appeared in Niko's hand it will still cancel the animation, but the helmet will immediately appear on Niko's head.

Hold R2 when you are getting on a motorcycle. You will drive once Niko has started the motorcycle, and "forget" to wear a helmet.