We conduct a deep mid-infrared (mid-IR) census of nine massive galaxy clusters at (0 < z < 1.3) with a total of ~1500 spectroscopically confirmed member galaxies using Spitzer/IRAC photometry and established mid-IR color selection techniques. Of the 949 cluster galaxies that are detected in at least three of the four IRAC channels at the >=3σ level, we identify 12 that host mid-IR-selected active galactic nuclei (IR-AGNs). To compare the IR-AGNs across our redshift range, we define two complete samples of cluster galaxies: (1) optically selected members with rest-frame V AB magnitude < - 21.5 and (2) mid-IR-selected members brighter than (M*3.6 + 0.5), i.e., essentially a stellar mass cut. In both samples, we measure f IR-AGN ~ 1% with a strong upper limit of ~3% at z < 1. This uniformly low IR-AGN fraction at z < 1 is surprising given that the fraction of 24 μm sources in the same galaxy clusters is observed to increase by about a factor of four from z ~ 0 to z ~ 1; this indicates that most of the detected 24 μm flux is due to star formation. Only in our single galaxy cluster at z = 1.24 is the IR-AGN fraction measurably higher at ~15% (all members; ~70% for late-types only). In agreement with recent studies, we find that the cluster IR-AGNs are predominantly hosted by late-type galaxies with blue optical colors, i.e., members with recent/ongoing star formation. The four brightest IR-AGNs are also X-ray sources; these IR+X-ray AGNs all lie outside the cluster core (R proj >~ 0.5 Mpc) and are hosted by highly morphologically disturbed members. Although our sample is limited, our results suggest that f IR-AGN in massive galaxy clusters is not strongly correlated with star formation at z < 1 and that IR-AGNs have a more prominent role at z >~ 1.