What specific mutations can cause the apoptosis mechanisms in a cell to malfunction? Are any such mutations 'reversible' , somehow or are they generally permanent? what kind of mutations can happen without disrupting the apoptosis mechanisms?
There are lots and lots of specific single mutations that cause problems with apoptosis.
Some of the most common are in P53, PTEN, MYC, APC, and KRAS. If you want a specific amino acid change, KRAS G12D is particularly potent. It works by disabling a regulatory domain and causing KRAS to be constitutively active.
These aren't "reversible" in a strict sense (except, I guess, for the struck-by-lightning-while-holding-a-winning-lottery-ticket level improbability of a second replication error in the same place back to the original base) but in many cases the cells will upregulate compensatory pathways or activate senescence pathways instead, rather than instantaneously becoming full-blown malignant.
As for mutations that don't cause disruptions to apoptosis,there are almost an infinite number, in that silent mutations or mutations to genes that aren't involved in apoptosis would count.
If you mean what kind of non-silent mutations can happen to genes that are important to apoptosis without causing problems, there are still a lot: swapping similar amino acids in non-active parts of proteins are unlikely to have huge effects.