how does a player with a red card get affected next game if they are eliminated

A player who gets a red card is usually suspended for the next match, so they cannot play if their team advances to the following game. If the team is already eliminated, there is no next match to miss, so the suspension just carries over to the player’s next official match with that team or competition, depending on the rules.

In soccer tournaments, a straight red or a second yellow often means at least a one-game ban, and if the suspension can’t be served in the tournament, it is carried to the team’s next official national match.

Simple example

• Player gets sent off in the last match of a group stage.
• Team is eliminated.
• The player does not miss another tournament game, because there isn’t one.
• If a ban still remains, it applies to the next official match the team plays.

Important note

If you meant a different sport or competition, the exact penalty can vary, but in soccer this is the general rule.

TL;DR

Red card = suspension. If the team is eliminated, the player usually serves that suspension in the next official match they are eligible for, not in a nonexistent “next game.”