Our game to decide the playoffs today.
 

Team A has 9 players. During an inning, Team A has a runner pull up injured but stays in the game. A couple innings later, the injured player ends up at second base. Team A inserts a Courtesy Runner for him. He is neither the pitcher or catcher of the previous inning. When Team A's coach is approached by Team B's coach about if he is being brought in to pitch or catch the next inning, they say yes. The inning is then over and the injured player who was courtesy run for proceeds to take the position of right field. Team B's coach approaches the home plate umpire and ask for an explanation and ruling. After meeting with the field umpire, they decide to allow this move and assess no penalty. What is Team B's course of action. The rules are Ohio High School Baseball Rules or NFHS.

Team B should have protested at the time the illegal substitution (that is what this is) was allowed by the umpires. A courtesy runner can only enter for the last player to play catcher or pitcher with the exception of the top of the 1st inning (3.1.1N.)

FED does allow a team with nine players to have a substitute runner (not a courtesy runner) when a player is injured and cannot run the bases. The substitute must be the most recent batter not on base.

And, the injured player can return to the game if / when he has sufficiently recovered.

That sounds to me (perhaps) like what happened.

The cases are in rule 4 somewhere -- whatever rule deals with playing with 8 players.

4-4-1f
NOTES
...If the offensive player must be substituted for after reaching base, the most recent batter not on base is allowed to run for that player.

This brings up an interesting question if protests are allowed...

If the outcome is legal (as this would be under the above cite) but the explanation is not (under 3.1.1N,) should the protest be upheld?

I would NOT uphold the protest.

And, to that point, the OP never said how the runner affected the game