It seems that, in general, irrationality or transcendental proofs of some "difficult" constant, like $e,\pi$ or $e^\pi$, relies on showing that there is an integer in $(0,1)$. But it seems there is no consistent way to reach this contradiction.
A kind of consistent proof of irrationality (the only one maybe) are the use of "Beukers Integrals" which can be used to show that these following numbers are irrational: $\ln 2, e, \pi^2, \zeta(2),\zeta(3) $. Basically, you need to construct a integral $I_n$, such that, $I_n = (a_n\xi+b_n)/d_n$, where $a_n,b_n,d_n$ are integers and $d_nI_n \to 0$ as $n$ grows larger, thus showing an integer between zero and one. However, I think this method has been milked to death and has reached its limitation.
Apery's proof for $\zeta(3)$ uses a fast convergent series for it. But it seems that this proof is "isolated", in a sense that it cannot be replicated to another constant. Looks like all irrationality proofs are "isolated" in this sense. They all lack similarities, except for the Beukers method mentioned.
Is there any specific math tool or a math field that is useful in studying or creating an irrationality proof?
For example, this paper includes some general and particular results in transcendental number theory with some proofs.
Certainly several areas are promising for finding a proof, for, say, the conjecture that $\zeta(3),\zeta(5),\zeta(7),\ldots$ are all irrational. In particular, number theoretical methods combined with combinatorics (asymptotic methods) might be useful. This is supported by the proof of Zudilin, who showed the following result with these methods:
Proposition: One of the numbers $ζ(5), ζ(7), ζ(9), ζ(11)$ is irrational.
He also has a very elementary proof for a weaker result here.