We have predicted extragalactic number counts and diffuse backgrounds at 25, 60, and 100-mu-m using new luminosity functions and improved spectral energy distribution density functions derived from IRAS observations of nearby galaxies. Galaxies at redshifts z < 3 that are like those in the local universe should produce a minimum diffuse background of 8.5 x 10(-3), 3.8 x 10(-2), and 0.13 MJy sr-1 at 25, 60, and 100-mu-m, respectively. Models with significant luminosity evolution predict backgrounds about a factor of 4 greater than this minimum. Strong luminosity evolution of infrared-bright galaxies can be excluded based on the predicted high fractions of sources at z > 0.5 that do not appear in the many redshift surveys of infrared-bright galaxies conducted to date. The minimum predicted background should be detectable by COBE at 100-mu-m and might be detected at 60-mu-m. COBE will probably not detect the 25-mu-m background. The next generation of space missions (e.g., ISO and SIRTF) will easily detect and characterize the evolution of infrared-bright star-forming galaxies.