Most of the time the person asking this question will answer like this: “Of course, the New Testament takes priority.” (In fact, the ones who ask that question are usually asking because they’re trying to make that point. Otherwise, it’s not really a question that needs to be asked.)
The problem is that the question as stated introduces a false dichotomy. Sometimes it is OT, sometimes it is NT, sometimes it is OT and NT (both-and rather than either-or).
Sometimes the NT does take priority. When the NT takes an OT promise or prophecy and applies it in a manner we wouldn’t expect, then certainly it makes sense to go with the NT. For example, Hebrews 8 takes the New Covenant passage of Jeremiah 31 and applies to the church.
The problem comes when one assumes that because a few OT passages are appropriated by the NT and applied to the church or to the Gentiles that all OT promises and prophecies must be so appropriated. For example, nowhere does the NT take the promise of Zechariah 14 and demonstrate its fulfillment in the church. Why then do we not take the OT passage at face value? In this case we have no reason to look for a fulfillment other than that expressed by the plain meaning of the passage. In this case, the OT takes priority because there is no NT change.
Sometimes, however, it is not “either-or,” we must either accept the NT interpretation or we must stay with the original meaning of the OT promise or prophecy, it is “both-and,” both NT interpretation and OT significance must be given their full value. For example, the Hebrews 8 passage mentioned above takes the New Covenant and applies it to the church. It does not, however, say that the Jeremiah 31 passage is now so fulfilled that there is no longer any further significance for ethnic Israel. Why then can we not see it as fulfilled in the church yet still awaiting its fulfillment for Israel? Many OT prophecies have near fulfillments and far fulfillments, why must the near fulfillment be limited to the OT era? Why cannot the “already-not yet” paradigm not be applied to prophecies such as this as well?
The only reason why a person doesn’t think it can be is because they don’t want it to be.