We prove two Lieb-Schultz-Mattis type theorems that apply to any translationally invariant and local fermionic d-dimensional lattice Hamiltonians for which fermion-number conservation is broken down to the conservation of fermion parity. We show that when the internal symmetry group G_f is realized locally (in a repeat unit cell of the lattice) by a nontrivial projective representation, then the ground state cannot be simultaneously nondegenerate, symmetric (with respect to lattice translations and G_f), and gapped. We also show that when the repeat unit cell hosts an odd number of Majorana degrees of freedom and the cardinality of the lattice is even, then the ground state cannot be simultaneously nondegenerate, gapped, and translation symmetric.