No. The set of primes can be whatever you want.
First, note that for infinitely presented groups, the torsion can be whatever you like: the torsion in the group
$*_i \mathbb{Z}/p_i$
is precisely the set of primes $p_i$, by standard facts about free products.
By Higman's Embedding Theorem, the above group can be embedded in a finitely presented group. More subtlely, this embedding doesn't introduce any new torsion---see, for instance, Theorem 2.5 of this preprint of Chiodo.