In my limited experience (which may soon be changed! :P), merely negating "fundamental" axioms does not yield strong in-system consequences. The word "merely" is doing some work here, though, since of course we can do more than simply add the negation of an axiom; e.g. Boffa's antifoundation axiom implies that every theory has a transitive model which is rather amusing, and similarly if we think of the axiom of determinacy as a strong anti-choice axiom then of course ZF+AD proves interesting new facts.
However, the resulting sysmtemssystems themselves may have surprising properties. In the particular case of extensionality, Scott showed that $\mathsf{ZF-Ext}$ (phraseddphrased using $\mathsf{Replacement}$ rather than $\mathsf{Collection}$) is equiconsistent with $\mathsf{Z}$, so adding extensionality to $\mathsf{ZF}$ results in a huge increase in consistency strength; one consequence of this (which I can't elaborate on, not being familiar with Scott's proof) is that there must be models of $\mathsf{ZF-Ext+\neg Ext}$ which are "much simpler to build/verify" than any models of $\mathsf{ZF}$. This doesn't contradict the previous paragraph, though, since this interesting result isn't a new theorem of $\mathsf{ZF-Ext+\neg Ext}$ but about $\mathsf{ZF-Ext+\neg Ext}$.
(If you shift from $\mathsf{ZF}$ to $\mathsf{NF}$, we get another example in Specker's theorem that $\mathsf{NFC}$ is inconsistent while $\mathsf{NFUC}$ - which is to say, $\mathsf{NFC-Ext}$ - is consistent.)