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][1].


  [1]: http://arxiv.org/abs/1107.1489