no they should not, and if they do, it may be any number of reasons. For one, check your grip and how much strength you use. Muscles up the arm tighten along with the grip. Notice what your upper traps are doing. I often watch how cyclist almost have their shoulders up towards their ears and tighten the entire upper body.
On the assumption that the bicycle is fitted well for you, I would check those things first and see whether that makes a difference.