We present a new, independent survey of the infrared emission for 161 Galactic supernova remnants using Skyflux image data from the Infrared Astronomical Satellite. We have derived new kinematic distances for 11 of the 44 positively or possibly identified remnants, using a new Galactic rotation curve with current IAU constants, R0 = 8.5 kpc and V0 = 220 km s-1. This new survey, which both complements and improves on the results of previous surveys, indicates that the infrared region is the equal of other spectral regimes for remnant detection. We used temperature-sensitive flux ratio images to help identify and define remnant emission structures and fluxes, particularly in complex regions. Of the 161 remnants examined, we found clear infrared emission from 35, with nine additional possible detections. Our detection rate is highest toward the Galactic anticenter, where source confusion is less severe. By fitting the observed fluxes to a two-temperature dust model with lambda--1.5 grain emissivities, we provide color-corrected fluxes in the four IRAS bands for every identified source. Planck-averaged grain opacities then yield remnant luminosities and dust mass estimates. Almost all young remnants (Tycho, Kepler, Cas A, Crab) have a significant 12 and 25-mu-m flux, while older remnants are brightest at 60 and 100-mu-m. With remnant radius as an age estimator, correlations with mass and temperature indicate that older remnants are both colder and more massive, as expected from decreasing shock velocity. No radius-luminosity correlation is seen. Good agreement generally exists between a remnant's nonthermal radio emission and its IR emission structure. Since a young remnant's dust is heated primarily by electron collisions in high-temperature plasmas, even better agreement can be seen between a remnant's IR and X-ray structure (e.g., the Cygnus Loop). However, the key advantage of far-IR observations is their ability to detect massive, slowly expanding, dusty shells, which often are weak in optical or nonthermal radio emissions. The main disadvantage of far-IR imaging is the moderate-to-severe source confusion in or near the Galactic plane owing to negligible interstellar extinction.