The 27-year-old Frenchman was sent off by referee Chris Foy in the 81st minute of the Premier League game at Carrow Road following an altercation with Canaries midfielder Bradley Johnson.
Johnson was also shown a straight red card after the pair shoved each other and came head-to-head, and Norwich appealed against his dismissal on Wednesday.
Newcastle manager Alan Pardew said after the game he would not appeal Remy's red card, but the club have now decided to try and get the decision overturned.
Pardew said on Tuesday: "There is probably just enough for the referee to send them off, but Johnson's reaction to the heads meeting, in terms that he got a headbutt when he did not, was the catalyst for them both to get sent off.
"It was his own fault, if he had not have done that then maybe the referee would have taken a more lenient view, but he made such a drama out of it, he should be ashamed of himself really.
"We won't appeal, because I think there was enough for Chris there, but once he has done that stupid reaction, the crowd went all mad and the referee has got no choice really."
Should Remy fail in his appeal, he will miss the Premier League games against Sunderland, Chelsea and Tottenham.