The striker scored a stoppage-time leveller to earn the Reds a 2-2 draw at Anfield on Sunday.
However, Lawrenson feels that the Uruguayan, whose performance was overshadowed after he aimed a bite at the Serbian international following an altercation in the penalty area, should face a long spell away from the game.
"What he did has absolutely no place in football and he is going to get - and deserves - an extremely lengthy ban," Lawrenson told BBC Sport. "A fine doesn't really matter with these lads because money is not an issue with them as they have got so much of it.
"I don't think you can argue against a lengthy ban, no matter who you support. You cannot bite people anywhere, let alone on a football field. It is the type of
thing you do when you are a baby.
"I've absolutely no idea what drives you to do what he did. The only one who can answer that is Luis Suarez."
Suarez has sinced issued an apology for his actions.