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The ex-England manager believes Martin's inflexibility contributed to his departures from Southampton and Rangers and warned it could lead to further job losses.
Allardyce said: "I wouldn’t give any advice to Russell Martin at Leicester, because I know he wouldn’t take it. It cost him his job at Southampton, it cost him his job at Rangers, and it’s going to cost him his job at Leicester.
"His beliefs are far away and beyond the capabilities of the players he’s using. The players probably don’t like it either."
Allardyce highlighted Martin's stint at Southampton as an example of his refusal to adapt. "When he went into the Premier League with Southampton, there was no adjustment to trying to get results, based on the purity of his beliefs in football."
He said that managers must be willing to change their style when necessary. "You live and die by your beliefs," he continued.
"Martin shouldn’t listen to me just because I say something, but when you keep losing your job because of your style of play, then what do you do?
"You need to stay in the game."
He cited Daniel Farke as an example of a manager willing to adapt. "You look at what Daniel Farke did at Leeds after coming back to the Premier League – they started to go direct through Dominic Calvert-Lewin, and it saved his job!"