We study excitons in suspended membranes of atomically thin WSe₂. We perform reflectance measurements to obtain the peak energies of the 1s, 2s, and 3s states of the A exciton in supported and suspended samples. From the experiments, we calculate the excitonic binding energies by employing the quantum electrostatic heterostructure model and the Rytova-Keldysh potential model. We find that the binding energy of the 1s exciton increases from about 0.3 eV (on the substrate) to above 0.4 eV (suspended) due to the reduced dielectric screening. We also exploit the tunability of the excitons in suspended samples via strain. By applying an air pressure of 40 psi to cause strain, we obtain reversible 0.15 eV redshift in the exciton resonance of a suspended 1L sample on a circular hole of 8 μm diameter. Interestingly, the linewidth of the 1s exciton decreases more than half from about 50 meV to 20 meV under 1.5% biaxial strain, due to the suppression of the intervalley exciton-phonon scattering. By making use of the strain-dependent optical signatures observed, we obtain the 2D elastic moduli of 1L and 2L WSe₂. Our results exemplify the use of suspended 2D materials as novel systems for fundamental studies and, strong and dynamic tuning of their optical properties.