We analyze the star formation properties of 16 infrared-selected, spectroscopically confirmed galaxy clusters at $1 < z < 1.5$ from the Spitzer/IRAC Shallow Cluster Survey (ISCS). We present new spectroscopic confirmation for six of these high-redshift clusters, five of which are at $z>1.35$. Using infrared luminosities measured with deep Spitzer/MIPS observations at 24 $mu$m, along with robust optical+IRAC photometric redshifts and SED-fitted stellar masses, we present the dust-obscured star-forming fractions, star formation rates and specific star formation rates in these clusters as functions of redshift and projected clustercentric radius. We find that $zsim 1.4$ represents a transition redshift for the ISCS sample, with clear evidence of an unquenched era of cluster star formation at earlier times. Beyond this redshift the fraction of star-forming cluster members increases monotonically toward the cluster centers. Indeed, the specific star formation rate in the cores of these distant clusters is consistent with field values at similar redshifts, indicating that at $z>1.4$ environment-dependent quenching had not yet been established in ISCS clusters. Combining these observations with complementary studies showing a rapid increase in the AGN fraction, a stochastic star formation history, and a major merging episode at the same epoch in this cluster sample, we suggest that the starburst activity is likely merger-driven and that the subsequent quenching is due to feedback from merger-fueled AGN. The totality of the evidence suggests we are witnessing the final quenching period that brings an end to the era of star formation in galaxy clusters and initiates the era of passive evolution.