We study the mid-infrared properties of 1315 spectroscopically confirmed members in eight massive (Mvirgtrsim 5 × 1014 Msun) galaxy clusters covering the redshift range from 0.02 to 0.83. The selected clusters all have deep Spitzer/MIPS 24 μm observations, Hubble and ground-based photometry, and extensive redshift catalogs. We observe for the first time an increase in the fraction of cluster galaxies with mid-infrared star formation rates higher than 5 Msun yr−1 from 3% at z = 0.02 to 13% at z = 0.83 (RP ≤ 1 Mpc). This increase is reproduced even when considering only the most massive members (M* ≥ 4 × 1010 Msun). The 24 μm observations reveal stronger evolution in the fraction of blue/star-forming cluster galaxies than in color-selected samples: the number of dusty, strongly star-forming cluster galaxies increases with redshift, and combining these with the optically defined Butcher-Oemler members [Δ (B − V) < − 0.2] doubles the total fraction of blue/star-forming galaxies in the inner Mpc of the clusters to ~23% at z = 0.83. These results, the first of our Spitzer/MIPS Infra-Red Cluster Survey (SMIRCS), support earlier studies indicating that the increase in star-forming members is driven by cluster assembly and galaxy infall, as is expected in the framework of hierarchical formation.