Question

When is it appropriate to use a biased opinion in a research paper

Answer 1

It isn't really. A research paper is about the research itself and where it leads. When it's about your opinion, you should try your maximum to deliver in a way that it is not biased. Of course you can add someone else's biased opinions as part of the research and discussed that and even compare to other biased opinions.

Answer 2

All sources will have some element of bias in them due to the fact that they are products of a specific person's (or specific people's) views. The best ways to deal with bias are 1) to acknowledge the bias and comment on it, 2) to include several sources with differing biases and discuss their differences, or 3) the analyze what the existence of the bias might indicate about the issue.