Since the New Covenant is the completion and unfolding of the Old Covenant (not its replacement), I think the Ark would have a place as a Christian relic...not simply a neat historical old thing. It was after all where the very center of the presence of God for centuries resided on earth.
Of course it depends on what you mean by "relic." Not being Roman Catholic, I don't buy that things associated with particular saints give off holy power (making one's prayers more effective, for example). However, even the Bible indicates that the bones of a prophet actually raised someone back to life (2 Kings 13:21)...so, maybe we Protestants should re-think some of this relic theology.
"It was after all where the very center of the presence of God for centuries resided on earth."
Just last year I was in a Bible Study and the book was talking about this. It seems odd to us (me at least!). But, you read it in the Bible that this is the "tabernacle" (dwelling place) for God. And at the end of Exodus after Moses gets everything (the Ark, incense, basin, tent, etc.) set up "Then the cloud covered the Tent of Meeting, and the glory of the Lord filled the tabernacle."
Of course that changed once Jesus came and died and rose and the Holy Spirit took over. I imagine the Ark (if it still is around) is nothing more than a beautiful yet glorified container of wonderful relics now.