The black hole images obtained with the Event Horizon Telescope (EHT) are expected to be variable at the dynamical timescale near their horizons. For the black hole at the center of the M87 galaxy, this timescale (5-61 days) is comparable to the 6 day extent of the 2017 EHT observations. Closure phases along baseline triangles are robust interferometric observables that are sensitive to the expected structural changes of the images but are free of station-based atmospheric and instrumental errors. We explored the day-to-day variability in closure-phase measurements on all six linearly independent nontrivial baseline triangles that can be formed from the 2017 observations. We showed that three triangles exhibit very low day-to-day variability, with a dispersion of similar to 3 degrees-5 degrees. The only triangles that exhibit substantially higher variability (similar to 90 degrees-180 degrees) are the ones with baselines that cross the visibility amplitude minima on the u-v plane, as expected from theoretical modeling. We used two sets of general relativistic magnetohydrodynamic simulations to explore the dependence of the predicted variability on various black hole and accretion-flow parameters. We found that changing the magnetic field configuration, electron temperature model, or black hole spin has a marginal effect on the model consistency with the observed level of variability. On the other hand, the most discriminating image characteristic of models is the fractional width of the bright ring of emission. Models that best reproduce the observed small level of variability are characterized by thin ring-like images with structures dominated by gravitational lensing effects and thus least affected by turbulence in the accreting plasmas.