We use deep near-infrared and submillimeter observations of three massive lensing cluster fields, A370, A851, and A2390, to determine the average submillimeter properties of a K'-selected sample. The 38 extremely red objects (EROs; I-K' > 4) with K' < 21.25 have a very significant error-weighted mean 850 m m flux of 1.58 +/- 0.13 mJy. The ERO contribution to the 850 μm background is (1.88 +/- 0.16) x 10(4) mJy deg(-2), or about half the background light. The 17 very red objects (VROs; 3.5 < I-K' < 4) are also significantly detected (1.32 +/- 0.19 mJy), bringing the combined VRO and ERO contribution to (2.59 +/- 0.19) x 10(4) mJy deg(-2). There is a substantial systematic uncertainty (about a factor of 2) in this value due to field- to- field variation, but even with this uncertainty it is clear that a large fraction of the 850 μm background arises from red objects. An analysis of the VRO and ERO number counts shows that half of the population's 850 m m light arises in objects with demagnified magnitudes K' < 20, and half in fainter objects. On the basis of the I-J versus J-K' colors of the galaxies, the bulk of the submillimeter signal appears to arise from the dusty starburst galaxies in the red object population rather than from the high- redshift elliptical galaxies.