Well, it's definitely possible. I'm no doctor, but there's a few ways I can see riding affecting it:
* The handlebar material or texture may be causing an allergic reaction / aggravating your hands
* Callouses developing on your hands may be somehow affecting it
* Sweating/lack of airflow from holding the bars would also possibly contribute to skin problems
* The weight/position of your hands is causing strain that affects your hands
Do you use / have you tried gloves? Personally I don't use them since I never go for super long rides, but sounds like it may be worth a try if you go for a quality model?