If Canada broke the treaty, in theory, the U.S. could lay claim to parts of Ontario and Quebec. And if America broke it, Canada could get parts of Maine, Michigan, and Wisconsin. So, basically, North American geography as we know it is contingent on this early 1800s treaty remaining in effect.
The podcast was from December -- an eternity ago in these interesting times -- and I don't know whether anything has changed since then.
https://99percentinvisible.org/episode/653-beyond-the-99-inv...
The podcast's transcript suggests that their source for this is https://www.cbc.ca/news/canada/british-columbia/peace-arch-u.... The person confidently claiming that any closure of the park would result in catastrophe is an immigration lawyer, not a historian:
"Saunders said the treaty stipulates there could not be any boundaries or physical barriers erected on the northern border of the U.S. — and if either side violated that treaty — the boundaries revert back to pre-treaty."
Since the Treaty of Ghent restored the pre-War of 1812 borders of both the United States and Canada, this doesn't make any sense. Canadian historian C. P. Stacey states that the period after the War of 1812 actually saw more border fortification than the years before (https://www.jstor.org/stable/1840618).
I mentioned that the idea that a 19th century treaty keeps the US-Canada border demilitarized was a good story, but I think the truth is an even better one: that the border is demilitarized because both countries know that they can trust their neighbor. Let's hope it stays that way.