As the IFAB Laws of the Game - Law 3: The players - Extra Persons on the Field of Play states:
The coach and other officials named on the team list (with the
exception of players or substitutes) are team officials. Anyone not
named on the team list as a player, substitute or team official is an
If a team official, substitute, substituted or sent off player or
outside agent enters the field of play the referee must:
- only stop play if there is interference with play
- have the person removed when play stops
- take appropriate disciplinary action
Usually, sent-off players will argue with referees after being sent off but eventually they will leave the field. What we see more frequently is some spectators (outside agents) running onto the field and being removed by security guards. Even if the guards don't explicitly ask permission from referee for removing the outside agents, they act according to the aforementioned law of the game.
The same would apply to the sent-off player, who will be removed by security guards in case if he doesn't want to leave the field, because the previously mentioned law applies not only to outside agents but also to sent-off players.
So, to answer your questions:
Can a referee forfeit the game if the player doesn't leave?
No, he'll just have the person removed with the help of security guards.
Does the referee just keep slapping the player with more red cards?
No, there's no point of showing multiple red cards, because the referee can just add the player's behavior to his protocol, so that the competition body could take appropriate measures later. This question has been already discussed here.