Question

What would happen if new cells stopped forming when we reached maturity while old cells never died off?

Answer 1

technically you would remain same for indefinite time, but would accumulate environmental damage and much higher rate

Answer 2

as cells get old they accumulate damages in their biomolecules thus affecting their function , normal functioning of the cell and on the whole the organism itself. Hence as the cells age in tissues , they undergo programmed cell death (apoptosis) and the lost cells are replaced by resident tissue adult stem cell proliferation and differentiation into the tissue cells. Now if the old cells never died off , their DNA would accumulate damages in the form of mutations which would result in high cancer incidence in most of your tissues thus disrupting organ function and homeostasis. So you may survive for a longer time , but it wont be a quality life that you will lead. This is what happens in aging , as you get old the chances of getting cancer , cognitive disabilities, neurodegenerative diseases increases because your cells function abnormally.