A search has been conducted for CO in 16 dwarf irregular galaxies selected to span a wide range of infrared luminosity and color. CO was detected in six dwarf galaxies spanning the entire range in optical color and IR luminosity. The dwarf galaxies extend by three orders of magnitude in luminosity the correlation between IR and CO luminosities found for spiral galaxies. For the dwarf galaxy sample alone, those galaxies with the least evidence for ongoing star formation have the lowest L(IR)/L(CO) ratios, while the active, blue dwarfs have the highest values of this ratio. This is interpreted as evidence for depleted values of this ratio. For galaxies with L(IR) between 100 million and a trillion solar luminosities, the full range of L(IR)/L(CO) values at each luminosity. The ratio of L(IR)/L(CO) in II Zw 40 is the highest of any galaxy in the present sample and is comparable to the values found in interacting galaxies. 40 references.