We discuss the effect of elastic deformations on the electronic properties of bilayer graphene membranes. Distortions of the lattice translate into fictitious gauge fields in the electronic Dirac Hamiltonian that are explicitly derived here for arbitrary elastic deformations, including in-plane as well as flexural (out-of-plane) distortions. We include gauge fields associated to intra- as well as interlayer hopping terms and discuss their effects on the electronic band structure and on the transport properties of suspended bilayer membranes. In particular, we consider the electron-phonon coupling induced by the fictitious gauge fields and analyze its contribution to the electrical resistivity. Of special interest is the appearance of a linear coupling for flexural modes, in stark contrast to the case of monolayer graphene. This new coupling channel is shown to dominate the temperature- dependent resistivity in suspended samples with low tension.