What happens when the last remaining players refuse to kill each other?

Yes, the ring will close completely and deal damage to players until a team is dead. You seem to take about 10 damage per tick rate of about 1.5 seconds.

There are plenty of examples on YouTube if you search along the lines of "apex legends heal off". I picked this example video because it's short and has no other unnecessary commentary, just to show you it is completely closed and the damage rate.

You will take about 10 damage per tick rate of about 1.5 seconds aa the answer above said.

However, it will be beyond unlikely that you will both die at the same time (if you have the same health).The ring will hit one player before another, this could be by 0.01 seconds. The one who goes in the storm first, even by such a gap, will die first.

Lets say the players both have same health and somehow both hit the ring at the exact same time, it will be likely that the winner will be chosen randomly.