We present X-ray observations of the "redback" eclipsing radio millisecond pulsar and candidate radio pulsar/X-ray binary transition object PSR J1723-2837. The X-ray emission from the system is predominantly non-thermal and exhibits pronounced variability as a function of orbital phase, with a factor of ~2 reduction in brightness around superior conjunction. Such temporal behavior appears to be a defining characteristic of this variety of peculiar millisecond pulsar binaries and is likely caused by a partial geometric occultation by the main-sequence-like companion of a shock within the binary. There is no indication of diffuse X-ray emission from a bow shock or pulsar wind nebula associated with the pulsar. We also report on a search for point source emission and $γ$-ray pulsations in Fermi Large Area Telescope data using a likelihood analysis and photon probability weighting. Although PSR J1723-2837 is consistent with being a $γ$-ray point source, due to the strong Galactic diffuse emission at its position a definitive association cannot be established. No statistically significant pulsations or modulation at the orbital period are detected. For a presumed source detection, the implied $γ$-ray luminosity is $\lesssim$5% of its spin-down power. This indicates that PSR J1723-2837 is either one of the least efficient $γ$-ray producing millisecond pulsars or, if the detection is spurious, the $γ$-ray emission pattern is not directed towards us. ; Comment: 10 pages, 6 figures; accepted for publication in the Astrophysical Journal