We make use of deep 1.2 mm continuum observations (12.7 mu Jy beam(-1) rms) of a 1 arcmin(2) region in the Hubble Ultra Deep Field to probe dust-enshrouded star formation from 330 Lyman-break galaxies spanning the redshift range z = 2-10 (to similar to 2-3 M-circle dot yr(-1) at 1 sigma over the entire range). Given the depth and area of ASPECS, we would expect to tentatively detect 35 galaxies, extrapolating the Meurer z similar to 0 IRX-beta relation to z >= 2 (assuming dust temperature T-d similar to 35 K). However, only six tentative detections are found at z greater than or similar to 2 in ASPECS, with just three at > 3 sigma. Subdividing our z = 2-10 galaxy samples according to stellar mass, UV luminosity, and UV-continuum slope and stacking the results, we find a significant detection only in the most massive (>10(9.75) M-circle dot) subsample, with an infrared excess (IRX = L-IR/L-UV) consistent with previous z similar to 2 results. However, the infrared excess we measure from our large selection of sub-L* (<10(9.75) M-circle dot) galaxies is 0.11(-0.32)(+0.42) +/- 0.34 (bootstrap and formal uncertainties) and 0.14(0.14)(+0.15) +/- 0.18 at z = 2-3 and z = 4-10, respectively, lying below even an IRX-beta relation for the Small Magellanic Cloud (95% confidence). These results demonstrate the relevance of stellar mass for predicting the IR luminosity of z greater than or similar to 2 galaxies. We find that the evolution of the IRX-stellar mass relationship depends on the evolution of the dust temperature. If the dust temperature increases monotonically with redshift (proportional to(1 + z)(0.32)) such that T-d similar to 44-50 K at z >= 4, current results are suggestive of little evolution in this relationship to z similar to 6. We use these results to revisit recent estimates of the z >= 3 star formation rate density.