I'd say probably not, but it would likely allow you to cancel your membership early (and receive paid fees back, pro-rated).
This is actually a nice hypothetical for that idiotic vision of replacing law and the court system with algorithms. It's extremely unlikely that the specific case would be foreseen in a contract. There is a continuous spectrum of such changes, and it's impossible to formulate any specific rule that would capture them all.
Example A: The gym changes from keys to plastic membership cards. Would this be a breach of contract? I think most everyone would agree that no, it isn't.
Example B: The gym requires whole-genome sequencing (once), then requires a drop of blood every time you enter to check your identity? Breach of contract? -> Obviously.
For any two such changes, you can probably come up with yet another example that's somewhere in between. The closer they get, the more often you will find people disagreeing, yes. But that just shows how justice is a constant conversation not easily set in stone.
As for the specific case: European law really doesn't like biometric data, and it's unlikely they can get away with it.
(the following is based only on my knowledge of German and Portuguese law)
BUT, ) if they do, the pro-rated refund is the most likely outcome. It works both ways, though: if you move away, they also cannot require you to keep paying fees. It's a concept loosely translated as a "cessation of the foundational requirements of the contract).