He does get extremely nervous before each Task in GoF, and often before important Quidditch matches. But I think you're right, he doesn't recognize/name it as "fear" per se, nor (as I recall) does he consider backing out of things once he's agreed to them.

Also, those two cases are both games -- artificial danger. People certainly get hurt at Quidditch and in the Tournament, but death isn't a primary risk. You're right, he doesn't react with that kind of anxiety when he thinks Sirius is after him.