Question

Are there any producers who don't do photosynthesis?

Answer 1

The answer is "yes." Organisms that live in harsh environments are organism that most likely cannot produce. Organisms that live in places such as: Deep in the oceans, deserts, jungles, tundra's, etc.. Which means they cannot get the resources they need to complete photosynthesis. 

Hope this helps!

Answer 2

Actually yes. You can find organisms, like bacteria, living in deep oceans, which do not have access to sunlight. There are cases in which they use thermal resources in order to produce energy. They are called chemoautotrophs. You can find them around deep ocean "smokers".