Biden could probably do an executive order to keep it up for a couple more weeks before the courts overturn it. Congress has control of the purse and they could try paying everyone's rent for a bit, but that would never pass and is also short term.

The US government has strong property laws built into the Constitution and no President or body of congress can indefinitely stop rent collection or evictions even if they wanted too. The government literally would collapse before something like an eviction moratorium became permanent.