Nunez was dismissed in the 57th minute for thrusting his head into the face of Palace defender Joachim Andersen, with Liverpool trailing to Wilfried Zaha's first half strike for Patrick Vieira's side.
Luis Diaz equalised for the hosts shortly after Nunez's dismissal but Liverpool could not find the winner.
The summer signing from Benfica became the first Liverpool player to be sent off on his home debut since Joe Cole against Arsenal in August 2010.
The Uruguay international will miss Liverpool's next three matches against Manchester United, Bournemouth and Newcastle United.
Suarez had his fair share of disciplinary problems during his time in England with Liverpool, and he has offered his international teammate some sage words.
"With what happened to him the other day, I was talking to him because he is just starting out and to make him aware that from now on, opponents will be seeking him out twice or three times more often," he told Telemundo.
"Even more so in England, he's hearing this from an idiot who makes mistakes and suffers for them, but falling down and getting up again made me stronger. Don’t give them any more chances, [or] it will only get worse."
"Darwin is someone who listens a lot. He is very smart that way. I think he will try to turn this situation around
"It is not a big deal, we have all made mistakes, we have all been sent off. The problem is that he has only just arrived over there and in England, to put it lightly, they make a big deal out of everything."
