Using the three-telescope IOTA interferometer on Mount Hopkins, we report results from the first near-infrared (l = 1.65 km) closure-phase survey of young stellar objects (YSOs). These closure phases allow us to unambiguously detect departures from centrosymmetry (i.e., skew) in the emission pattern from YSO disks on the scale of 64 mas, expected from generic "flared disk" models. Six of 14 targets showed small, yet statistically significant nonzero closure phases, with largest values from the young binary system MWC 361-A and the (pre main-sequence?) Be star HD 45677. Our observations are quite sensitive to the vertical structure of the inner disk, and we confront the predictions of the "puffed-up inner wall" models of Dullemond, Dominik, & Natta (DDN). Our data support disk models with curved inner rims because the expected emission appears symmetrically distributed around the star over a wide range of inclination angles. In contrast, our results are incompatible with the models possessing vertical inner walls because they predict extreme skewness (i.e., large closure phases) from the near-IR disk emission that is not seen in our data. In addition, we also present the discovery of mysterious H-band "halos" (65%-10% of light on scales 0."01-0."50) around a few objects, a preliminary "parametric imaging" study for HD 45677, and the first astrometric orbit for the young binary MWC 361-A.