Recent observations of Type Ia supernovae (SNe) at redshifts 0 < z < 1 reveal a progressive dimming that has been interpreted as evidence for a cosmological constant of Omega(A) similar to 0.7. An alternative explanation of the SN results is an open universe with Omega(A) = 0 and the presence of greater than or similar to 0.1 mu m dust grains with a mass density of Omega(dust) similar to a few x 10(-5) in the intergalactic (IG) medium. The same dust that dims the SNe absorbs the cosmic UV/optical background radiation around similar to 1 mu m, and reemits it at far-infrared (FIR) wavelengths. Here we compare the FIR emission from TG dust with observations of the cosmic microwave (CMB) and cosmic far-infrared backgrounds (FIRB) by the DIRBE/FIRAS instruments. We find that the emission would not lead to measurable distortion of the CMB, but would represent a substantial fraction (greater than or similar to 75%) of the measured value of the FIRE in the 300-1000 mu m range. This contribution would be marginally consistent with the present unresolved fraction of the observed FIRE in an open universe. However, we find that IG dust probably could not reconcile the standard Omega = 1 CDM model with the SN observations, even if the necessary quantity of dust existed. Future observations, capable of reliably resolving the FIRE to a flux limit of similar to 0.5 mJy, along with a more precise measure of the coarse-grained FIRE, will provide a definitive test of the IG dust hypothesis in all cosmologies.