We present Spitzer MIPS 24 mu m observations of sixteen 0.4 < z < 0.8 galaxy clusters drawn from the ESO Distant Cluster Survey. This is the first large 24 mu m survey of clusters at intermediate redshift. The depth of our imaging corresponds to a total IR luminosity of 8 x 10(10) L-circle dot, just below the luminosity of luminous infrared galaxies (LIRGs), and 6(-1)(+1)% of M-v < -19 cluster members show 24 mu m emission at or above this level. We compare with a large sample of coeval field galaxies and find that while the fraction of cluster LIRGs lies significantly below that of the field, the IR luminosities of the field and cluster galaxies are consistent. However, the stellar masses of the EDisCS LIRGs are systematically higher than those of the field LIRGs. A comparison with optical data reveals that similar to 80% of cluster LIRGs are blue and the remaining 20% lie on the red sequence. Of LIRGs with optical spectra, 88(-5)(+5)% show [O II] emission with EW([O II]) 5 angstrom, and similar to 75% exhibit optical signatures of dusty starbursts. On average, the fraction of cluster LIRGs increases with projected clustercentric radius but remains systematically lower than the field fraction over the area probed (<1.5 x R-200). The amount of obscured star formation declines significantly over the 2.4 Gyr interval spanned by the EDisCS sample, and the rate of decline is the same for the cluster and field populations. Our results are consistent with an exponentially declining LIRG fraction, with the decline in the field delayed by similar to 1 Gyr relative to the clusters.