I thought I would add more detail to my overly simplistic and perhaps somewhat misleading account of the philosophical criticism of Tarski’s theory of truth.
There is criticism of Tarski’s T-Scheme (and recursive definition) and there is criticism of Tarski’s philosophical view that our intuitive notion of truth is self-contradictory.
Historically, the earliest criticism of Tarski’s T-Scheme has its roots in the writing of the polymath F.P. Ramsey and goes under the name deflationism. Ramsey identified the equivalence principle which is present in Tarski’s T-Scheme as leading to the view that the idea of a truth predicate is redundant. Thus, when we say “snow is white” is equivalent to saying “it is true that snow is white”, the difference between them is merely stylistic and the truth predicate is redundant.
Other deflationists were quick to follow. For example, P.F. Strawson argued that while “snow is white” does express a proposition, “it is true that snow is white” does not express a proposition about a proposition, as Tarski would contend. Rather, it should be treated as a type of endorsement of the proposition “snow is white”.
Next to criticize Tarski was W.V.O Quine with a view called quotationalism. For Quine, when Tarski intends
”Snow is white” is true if and only if snow is white.
the quotation marks turn the sentence Snow is white into something noun-like. This leads to a type of circularity since adding the truth predicate gives us back a sentence whose truth we understand as well or poorly as we understood the original sentence itself. Quine gives his own analysis of truth predicates.
Another philosophical issue with Tarski’s T-Scheme goes under the name of indeterminacy. Problems here include, for example, problems of vagueness. Natural languages include many vague predicates - e.g., is bald, is tall, is a heap. This makes it hard, if not impossible to determine the truth of the axiom:
”Augustin-Louis Cauchy was bald” is true if and only if Augustin-Louis Cauchy was bald.
This is a taste of some of the early criticisms of Tarski’s philosophical theory. The Stanford Encyclopedia of Philosophy provides a detailed account.
Regarding mathematical logic and the status of Tarski's semantic definition of truth and his theorem that there is no 1-ary relation that identifies the true formulae of formal arithmetic, these are fundamental to first-order logic. I'm certainly not aware of any controversy amongst mathematical logicians.
Regarding Tarski's philosophical ideas it is important to keep in mind that no philosophical theory is without controversy. Any metaphysical theory will have to confront the realist/anti-realist divide. In the case of Tarski's theory of truth, a "realist" will believe that a proposition is either true of false regardless of whether it is decidable, while an "anti-realist" will believe that an undecidable proposition has no truth value.
At first glance Tarski's semantic definition of truth may appear to suggest that Tarski was a realist, but this is not clear because, after all, this is philosophy so the idea of being clear is out of the question. Regardless of whether Tarski was a realist or anti-realist his theory would be subject to criticism from the other side.
TL/DR (and probably not relevant)
I'm not sure how well acquainted you are with philosophy, and I'm certainly no expert, but it may be helpful to keep in mind that Tarski's use of the word semantic is an archaic usage, prevalent amongst positivist philosophers in the 1930s. (Tarski's T-Schema make clear that Tarski had a positivist attitude.) The positivists used a trio of terms: syntactic, semantic, and pragmatic. Since this answer is so far unreferenced, I'll slip in:
A notion was called syntactic if it pertains to relations among words, semantic if it pertains to relations among words and things, and pragmatic if it pertains to relations among words, things, and people.
[Source: Truth, by Burgess & Burgess, Princeton Press, 2011].
So Tarski's use of the word semantic had nothing to do with its current use in linguistics as "meaning".
"Snow is white" is true iff snow is white
is semantic because it involves words and a thing (snow), but no people.
Since Tarski was a mathematician doing philosophy it is difficult to say whether he was a realist or anti-realist. As a mathematician he was most likely a realist (i.e., a Platonist) while as a philosopher he should be an anti-realist. So here we have another source of controversy: the mathematicians will tend to poo-poo the philosophers while the philosophers will tend to laugh at the mathematicians naivety.