Shamir's Secret Sharing on non-prime GF
You appear to be confused by the sentence in the answer:


  If you use simple modular reduction they won't work (e.g. you can't just compute the operations mod 256).


This sentence is true, but doesn't mean what you think it does.

It doesn't mean that you can't compute inverses in $GF(256)$; instead, what it means is that you can't compute the addition and multiplication as $A + B \bmod 256$ and $A \times B \bmod 256$; other algorithms need to be used.

It happens that addition is easy; you can do it by doing a bitwise exclusive or; in C, that's A ^ B

Multiplication and inverses are a bit trickier; however, I can point you to the answer here that addresses how to do with with a few table lookups