UPDATE: It is well known that for any irrational number $\alpha$, given a $\epsilon > 0$, the inequation :$$|\alpha -\frac{p}{q}|< \frac{1}{q^{2+\epsilon}}$$, if $\alpha$ is algebraic, the inequation has finite solutions, if the inequation has infinite solutions, $\alpha$ is transcendental. This is what called Roth's Theorem.
Given an irrational function, and suppose it is approximated by polynomials, is there any theorem like the Roth's Theorem?
I suppose by "irrational function" you mean a function that is not a rational function. But maybe what you are really interested in, analogous in some way to Roth's theorem, is the distinction between algebraic functions and transcendental functions. And one way (not the only way) of approximating functions by polynomials is Taylor series. So, consider this. If $f(z)$ is an algebraic function that is not a polynomial, then its Taylor series must have finite radius of convergence. In other words, an entire function that is algebraic must be a polynomial. See my answer here for a proof.