3 players are involved in a hand pre-flop. All other players fold in turn.

Player 1 has $800, Player 2 has $1000, Player 3 has $395

Player 1 opens for $20, Player 2 raises to $60, Player 3 calls. Player 1 re-raises to $260, Player 2 calls, Player 3 re-raises all-in, Player 1 attempts to re-raise all in.

At this point a discussion occurs about whether or not the raise all-in from Player 3 is legitimate since it is not twice the amount of Player 1's re-raise and therefore the action is closed and all the other 2 players can do is call. The floor is called and rules that the action is closed and only a call is in order.

Is Player 1 allowed to re-raise all-in or is the action, in fact, closed and only a call is in order?

The raise all-in is legitimate since you can always go all-in in a NL game.

That said, his raise didn’t reopen the betting to players 1 and 2. So they can call or fold but not raise.

Player 3's all in would have needed to reach 460 in order to allow Player 1 to re-raise again. The reason there likely was a debate is due to how limit hold'em works which reopens the betting if the all-in is 50% or more of a full raise.