If enough people FIREd, yes I agree with you that'd start to drive up salaries for the folks who remained in the workforce. In order for that to be noticeable, I suspect that'd require an even greater adoption than the level of adoption required for us to influence the national savings rate.
Right now americans as a whole have a savings rate of ~2%. So even one person out of 100 goes from saving 2% to saving 60%, that'd increase our average savings rate to ~2.6%.
The other issue is that folks with higher incomes will work fewer years before FIREing, so if enough of us were dropping out of the labor force to make a difference, the effect would be felt most strongly for occupations which already have higher pay and less strongly for occupations with low pay. (If fact, if you believe the folks who talk about "barista FIRE" you might actually see an INCREASE in the labor force for causal unskilled labor.)