Suarez has been barred from all football-related activity by FIFA for four months after biting Giorgio Chiellini in Uruguay's 1-0 World Cup win over Italy on Tuesday.
In addition, the striker faces a nine-match international suspension, having previously received sanctions for similar offences when representing Ajax and Liverpool.
The 27-year-old was handed a 10-match ban after biting Chelsea defender Branislav Ivanovic last April and Lawrenson, who enjoyed great success in a seven-year spell with Liverpool from 1981 to 1988, feels it is time for the Merseyside club to part company with a player that has reportedly attracted the interest of Real Madrid and Barcelona.
"It's just a case of 'here we go again', isn't it? I don't really think that the punishment fits the crime," Lawrenson told BBC Radio 5 Live.
"How do you explain to your lad who's a football fan exactly what Luis Suarez keeps doing?
"He is now obviously a persistent offender and I thought actually the ban would have been much harsher.
"What does Liverpool owner John Henry do now? I would think he wants to wash his hands of Luis Suarez. Henry will try and get the best possible deal for Liverpool. Somebody will buy him. It is football when all is considered."
Lawrenson's former central defensive partner Alan Hansen feels Liverpool can give Suarez one more chance.
"I think the punishment's fair," Hansen said.
"Obviously, the worrying thing for any Liverpool supporter was that there was very little or no provocation, and the question is: Will he do it again?
"Liverpool have got to draw a line under this and say: 'One more indiscretion and it's over.'"
