We study theoretically the effects of an anisotropic elastic strain on the exciton energy spectrum fine structure and optical selection rules in atomically thin crystals based on transition-metal dichalcogenides. The presence of strain breaks the chiral selection rules at the K points of the Brillouin zone and makes optical transitions linearly polarized. The orientation of the induced linear polarization is related to the main axes of the strain tensor. Elastic strain provides an additive contribution to the exciton fine structure splitting, in agreement with experimental evidence obtained from the uniaxially strained monolayer. The applied strain also induces momentum-dependent Zeeman splitting. Depending on the strain orientation and magnitude, Dirac points with a linear dispersion can be formed in the exciton energy spectrum. We provide a symmetry analysis of the strain effects and develop a microscopic theory for all relevant strain-induced contributions to the exciton fine structure Hamiltonian.