I'm interested in doing computations with certain noncommutative rings, most of which involve taking derived tensor products. Does anyone know of a computer algebra package which will find projective resolutions of complexes of modules over a finitedimensional noncommutative ring, tensor with a bimodule, and do it all over again?

I don't know whether Magma can handle all you ask for, but if I remember my coding for Magma correctly, at least the projective resolutions of modules over a noncommutative ring should be covered by that  for nice enough noncommutative rings. It's all been developed there as part of Jon F. Carlson's work on computing group cohomology rings. If there is a system that does all you ask for, and does it efficiently, it is probably been written in connection to a group cohomology computation effort  which narrows the candidates down significantly: Magma and GAP do group cohomology rings, and SAGE now with the work of Simon King and David Green. In contrast, I'm reasonably certain that Macaulay only does commutative things, and Singular doesn't have resolutions as a naturally occuring object at all. Bergman might be able to deal with what you ask for, though. To conclude: I'd recommend you to take a look at the homological algebra modules in Magma, GAP, SAGE and Bergman  I'd be highly surprised to see any other packages deal with the case you describe, and I'm not entirely convinced either of these do it well either. 

