Yes, although the woman does provide her own power for a spin the man provides some impetus, as a signal to spin or (especially) keep spinning, so the woman has to have enough tension in her arm to resist this and keep her hand in position relative to her head. But I suppose if it's done right it feels so smooth that it doesn't really feel like "tension" at all.
This business of arm tension gets right to the heart of partner dancing. It's all about cooperation and communication, and in dancing, as in life, some people are more cooperative and communicative than others. The only general principle I can come up with at the moment is - if it feels right, it is right.
It's obviously not enough to say, as some teachers do, "keep some tension in your arms". I think that's the root of the problem for some people, they take it as the complete and literal truth.