Just as Moses lifted up the snake in the desert, so the Son of Man must be lifted up, that everyone who believes in him may have eternal life.
Here Jesus uses an Old Testament reference to illustrate the nature of his coming death. That is, he uses the story of the bronze snake as an analogy. Jesus did not say that his own death was the true meaning of that incident. Not every reference to the Old Testament in the New is aimed at telling us the "real" meaning of the text. So the next question when approaching the use of the Old Testament in the New is: is the author making an assertion about the meaning of the text which is being referenced? In Jesus' case above, no he was not. It was an analogy, a reference. In other cases (and the Apostle Paul could provide many examples), the author is making an assertion about the meaning of the text. In those cases, it is an exposition. The difficult task is of course determining which is which. There are passages where analogy and exposition can both serve the same rhetorical function. For example, Matthew's infamous use of Hosea:
So [Joseph] got up, took the child and his mother during the night and left for Egypt, where he stayed until the death of Herod. And so was fulfilled what the Lord had said through the prophet: "Out of Egypt I called my son."
Scandal! I would have certainly failed a theology essay where I used a verse so flagrantly out-of-context to prove my point. Hosea is talking about Israel, clearly. Yet it seems that Matthew is saying that Hosea was really talking about Jesus. But is that really the case? "Fulfillment" language comes with its own debate, but allow me to suggest this: Matthew here is not making an assertion about the meaning of Hosea's words; rather he is pointing out how God's providence worked in a way parallel to another time in Israel's history. Furthermore, I believe he is using this parallel to "read in" the context of Hosea 11, which talks about Israel's rebellion and God's forbearance and compassion. In other words, Jesus' return from Egypt is an illustration of the core message of the gospel.
The use of the Old in the New is not always simple. That being said, I think the analogy/exposition dichotomy provides an important tool for forming a normative hermeneutic. To say what a passage means (the task of hermeneutics) is not always always the sole purpose of OT references in the NT. The rhetorical force of the quotation (which is always present) must also be considered. I think that a confusion between these two types of analysis (hermeneutics v. rhetorical analysis) has lead to a lot of confused "hermeneutical" systems for understanding the OT in the NT - namely midrash, pesher, sensus plenior, double-fulfillment, and the lot. Those are primarily describing the rhetorical force of OT quotations in the NT while not actually evaluating how the New Testament authors understood the meaning of the text. Evaluating their understanding, however, is the goal of the search for a normative hermeneutic. We can see that the New Testament authors used the Old Testament in a variety of ways. The greatest value in studying their usage is perhaps in forming a normative framework for how the scriptures may be employed rhetorically in sermons and theological writings. I suggest that such a framework would be relatively broad, based on the examples in scripture. Still, this task is distinct (though not unrelated) from the task of forming a normative hermeneutic.