Millimeter wave Very Long Baseline Interferometry (mm-VLBI) provides access to the emission region surrounding Sagittarius A*, the supermassive black hole at the center of the Milky Way, on sub-horizon scales. Recently, a closure phase of 0+-40 degrees was reported on a triangle of Earth-sized baselines (SMT-CARMA-JCMT) representing a new constraint upon the structure and orientation of the emission region, independent from those provided by the previously measured 1.3mm-VLBI visibility amplitudes alone. Here, we compare this to the closure phases associated with a class of physically motivated, radiatively inefficient accretion flow models, and present predictions for future mm-VLBI experiments with the developing Event Horizon Telescope (EHT). We find that the accretion flow models are capable of producing a wide variety of closure phases on the SMT-CARMA-JCMT triangle, and thus not all models are consistent with the recent observations. However, those models that reproduce the 1.3mm-VLBI visibility amplitudes overwhelmingly have SMT-CARMA-JCMT closure phases between +-30 degrees, and are therefore broadly consistent with all current mm-VLBI observations. Improving station sensitivity by factors of a few, achievable by increases in bandwidth and phasing together multiple antennas at individual sites, should result in physically relevant additional constraints upon the model parameters and eliminate the current 180 degree ambiguity on the source orientation. When additional stations are included, closure phases of order 45--90 degrees are typical. In all cases the EHT will be able to measure these with sufficient precision to produce dramatic improvements in the constraints upon the spin of Sgr A*.
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