It would depend on the condition of the padding. If it's the foam, "Grab-On" kind, I say take it off. I used to use that. It broke down in the course of a year or so and lost most of its ability to cushion. Nowadays there are gel pads you can put on under your bar tape, and there is bar tape that is called gel that is pretty soft. One thing to keep in mind: there's no law that says you can only use one thickness of bar tape. I've used the gel pads plus two layers of tape. My bars look awfullly fat, but they're soft and I don't get numb hands like I used to.

Hardcore riders undoubtedly think I'm a Fred, for this and other reasons. What, me worry?