No they do not have to increase for an economy to function.
In fact, in the US during the period between 1975 - 2010 real wages tended to approximately stagnate as you can see figure published in the Scientific American, Nov 2018, p 61 (note I obtained the figure from this older answer).
Now to be fair, there are some economists who will dispute this because it is hard to measure real wages and control for quality improvements of goods and services people buy with their wages, but it is generally agreed taht during 1975-2010 US real wages were either stagnant or if growing only very slowly.
Consequently, if you want to know what happens to an economy if there is no real wage growth over period of time just look at the US (1975-2010).
Clearly economy can function with stagnant real wages (it could function even with declining ones, as the graph shows there were periods of time where real wages even declined a bit).
More generally, virtually any macro model of an economy that we use would work and not break down with zero growth in real wages (see Romer Advanced Macroeconomics for an overview).
There is also no reason to think growth of real wages makes economy function less and vice versa (especially not in long run, in short run it is actually growth of real wages that can cause frictions if wages are sticky).
Whether this is not good or good enough or bad depends on your personal moral/political values and it is outside of realm of economics to ascertain.