In reply to john arran:
FYI, I have been a "chalk free" climber for about 8 yrs, regularly leading without chalk including up to F6b, HVS5a, single multi in the last year or so, though I do mostly climb about F6a/VS. Id actually say my finger strength/muscle and hand technique especially in jamming/use of balancy flatter holds has got much better since not using chalk. I also have learnt to feel around much more on (and in) holds, look for the best hand position, optimise more. While I havent gone up a grade, I do feel more solid with my hands and I have led routes since without chalk that I didnt on-site lead before with chalk, so not carrying chalk hasnt stopped that particular progression of mine and if anything helped it.
But sure my level is not at all high level, so really Id say it depends what grade you climb (or want to) if you are thinking of going chalk free. What I was saying above about hand technight improvements might not be relevant to grades like F6c/UK 5c onwards where climbers probably do that instinctively and hand hold options and margins are no doubt exponencially much tigthter anyway. So perhaps not using chalk might be detrimental at a higher level.
But if it is for health reasons, not using chalk would definitely be one of the options, as in I would swap high level leading with not having a skin condition for example, if chalk was causing it. Though you´´ll get chalky hands on a lot of routes anyway, whether you carry it or not.