Sources: Jacob Neusner's Introduction to Rabbinic Literature, and Judith Hauptman's "Mishnah As a Response to 'Tosefta,'" which is in Specific Problems in Rabbinic Literature.
What influenced what? That seems to be where Neusner and Hauptman disagree. According to Neusner, the Tosefta was like an early version of the Talmud in that it served as a commentary on the Mishnah. Hauptman, however, contends that the final form of the Mishnah came after the Tosefta.
Both use a similar argument for their positions. According to Neusner, there are parts of the Tosefta that make absolutely no sense without the context of the Mishnah, so the Tosefta was designed to supplement the Mishnah. Consequently, the Mishnah came first. According to Hauptman, however, there are parts of the Mishnah that make absolutely no sense without the details of the Tosefta, so the compiler of the Mishnah must have assumed that his audience already knew the Tosefta.
Personally, I would need an encyclopedic knowledge of both documents to interact with these scholars' claims. What I find interesting is Hauptman's argument that the detailed Tosefta had to precede the not-so-detailed Mishnah. She acknowledges that people can take an opposite track, by saying that the Tosefta is more detailed because it tries to clarify and explain the Mishnah. But she doesn't go that route.
This is relevant to Fishbane, who believes that biblical authors clarified and sought to explain other biblical texts. As far as he's concerned, the texts that seek to clarify are later than the texts that are supposedly being clarified. But is that necessarily the case? Hauptman's model of the Mishnah and Tosefta screams "not always!"
I'm reminded of something N.T. Wright said in a class I took with him on the resurrection of Jesus. Many New Testament scholars maintain that Matthew and Luke interpreted Mark and Q (a source of Jesus' sayings). After all, Mark is bare-bones on Jesus' life and activity! But N.T. Wright asked why we should assume that model. Maybe Mark was intended to be a condensed version of the other two Gospels! Can we be dogmatic about what source preceded what?