One of the tests for Fair Use in the US, as I understand it, would be whether the archived work "competes" with the original.
If people start going to IA instead to read the news, the newspaper might have a claim. But if they're doing it to get around paywalls, or purely for archival/historical/research purposes, that may be allowed.
But the reality is such decisions are subjective and will be up to whatever judge happens to get such a case in front of them if this is challenged.