Loss of feeling,tingling or burning pains in the arms or in the hands - i agree with Markharf that it can indeed in many cases be the onset of problems in the neck area - often started by damage, wear and tear and very often bad posture - which makes this much worse. (if you do not think your posture is bad try going to a physio, yoga or pilates specialist who will quickly assess you for this and it really makes a huge difference. The weight loading on the spine through bad posture is massive and can cause huge problems. All of this can be made worse when riding a bike - both through posture and tension.

A good physio and maybe some light pilates exercises may help but in my case - where it was chronic and disabling after an injury losing the use of one arm to nerve compression in the spine in the end - i underwent spinal surgery after ten years of ever increasing problems. ( I now have spinal fusion both in my neck and lumbar region (top and bottom of spine) and can now ride (on the road) a bike again. I also have full use of the arm and hand with no pain or tingling with some minor limitations to what i can do so as not to aggrevate the repair.

I would say along with religiously doing pilates exercise to keep the remaining area of spine mobile i have largely recovered and have very few problems providing i am sensible about what i am doing.

Don't let it get to that sort it early if you can.

Pongo is right that restrictive clothing also makes a difference but i would check out the medical side if this is a recent change. - posture, exercise and then if it gets worse a consider talking with a phyio and maybe a neuro specialist.