How to prove that expessions like $\sqrt{93+63\sqrt{85}} - \sqrt{143} \notin \Bbb{Z}$?

In this example, if $\sqrt{93+63\sqrt{85}}-\sqrt{143}=t\in\Bbb Q$, then $$93+63\sqrt{85}=(t+\sqrt{143})^2$$ but that implies that $\Bbb Q(\sqrt{85})=\Bbb Q(\sqrt{143})$ which is false. So, not only is your expression not an integer, it's not rational either.

This sort of argument will work for most expressions like $\sqrt{a+\sqrt b}-\sqrt c$ but not for all possible "rooty" expressions.

A more general method that works for these type of problems is to first find the minimal polynomial of the root expression (this is the polynomial of which the expression is solution and it's also of the lowest possible degree). There is an algorithm for this and you can use wolframalpha. This only works for algebraic numbers however.

For your example the minimal polynomial is $$x^8-944x^6-446946x^4-455778560x^2+112134658225$$

By the integral root theorem the divisors of $112134658225$ are the only possible integer solutions to this polynomial. The divisors are: $1,5,25,66973,334865,1674325,...$ and they get bigger and bigger. We also know that the root expression is solution to this polynomial and that it's close to $14$. However none of the divisors is anywhere close to $14$ so our root expression must be irrational solution.

This method only works for when the leading coefficient is $1$ but can be generalised even further with the rational root theorem.

I also recommend watching mathologers video here : https://www.youtube.com/watch?v=D6AFxJdJYW4 which more or less should answer some other questions you may have.