I'm pretty darn sure that the Tillamook Head Trail is strictly a hiking trail. And if for some reason they did allow bikes, this would be a moderate level mountain bike trail. I hiked it years ago, and there are some steep/rough sections. So, to answer your question, probably not appropriate.

Besides the Promenade in Seaside, there's not any multi-use paths that I know about in that area. Someone else can correct me if I'm wrong.