Once when I was a gabbai in a very small jewish community, we had two cohanim: one was our Rabbi and usually I gave him an aliya l'torah -called him up for a torah reading. The other was secular and had several young children and was usually busy with them during services. Because of that I usually tried to avoid giving him mitzvot that restricted his movement for a long time (such as hagba'ah which would require him to sit at the bima), or long p'tichot which would require him to stand at the aron. This was at his request.
I once asked him whether he would be able to do the peticha -open the aron hakodesh at hanoten teshua -a prayer for welfare the state of israel. He told me that I shouldn't ask a cohen, because a cohen is duty-bound to serve and if asked is always required to do what is asked in shul, i.e. forbidden to say no. I never heard of such a mitzva, but given that there are enough mitzvot I've never heard of before I decided not to argue. At the time I decided not to ask the rabbi because if the cohen was wrong I didn't want our rabbi to know automatically who made the original statement, since there were only two cohanim in our community. I left it as it is.
Now several years have passed and I still wonder: Was the cohen right? I've tried searching but found nothing. Is there any source that states that if (specifically) a cohen is asked whether he can be called for a mitzva during service, that he, because he is a cohen, is forbidden to say no?