By all accounts, Gödel rarely uttered a word when he attended the logical positivist's meetings in Vienna. So it is not clear if Gödel was originally opposed to logical positivism even if his opposition became absolute as time passed.
The refutation of logical positivism began when Russell pulled the rug out from under Frege with his famous paradox. We still have the polite but lethal letter Russell sent to Frege in 1902 :
Dear Colleague, ... I find myself in complete agreement with you in all essentials... I find in your work discussions and distinctions... one seeks in vain in the work of other logicians. There is just one point in which I have discovered a difficulty. ...
Hilbert's response was to introduce formalism in an attempt to bypass Russell's paradox. Gödel took up Hilbert's monumental programme by attempting to see whether one could prove the consistency and completeness of a formal axiomatic system for mathematical analysis. Of course, he discovered that neither was possible.
What Gödel first discovered was that mathematical truth cannot in principle be confined to a formal system - i.e., truth is not reducible to proof; syntax cannot supplant semantics; intuition cannot be dispensed with in mathematics, indeed, even in arithmetic. This was first metaphorical nail in the logical positivist coffin.
The second nail in the coffin of logical positivism came when Gödel demonstrated that if a given system of axioms for arithmetic were in fact consistent, then it could not be proved consistent by the system itself. In other words, only an inconsistent formal system can prove its own consistency.
The logical positivists did not go down without a fight. By all accounts, Hilbert's first response to Gödel's results was anger followed by denial. He became the first, but by no means the last, to propose an anti-Gödel principle; an ad hoc principle to be appended to formal mathematics simply to block the application of Gödel's theorems. Gödel was apparently genuinely irritated by this. Hilbert's idea was to append a new rule of deductive inference that would allow for the employment of infinitely many premises. Gödel was quick to point out that this would violate the very idea of a formal system. The cure would kill the patient.
Few of the logical positivists, or philosophers in general, appear to have initially fully understood Gödel's results - the exception being von Neumann. Gödel simply stopped responding to Zermelo's frequent letters. Carnap too, the first to hear Gödel's result, appears to have failed to comprehend its implications for some time. Gödel famously rubbished Wittgenstein's criticisms as trivial. And a host of others.
So although it is unclear whether Gödel initially opposed logical positivism, it is clear that his results spelt its demise and that he well understood it.