Have any natural complexity classes been proven not to be closed under complement?
There may be no such natural class, i.e. all natural classes are closed under complement.

Despite that $\mathrm{DTIME(n)\neq NTIME(n)}$ was proven, it is still possible that $\mathrm{NTIME(n^k)=co-NTIME(n^k)}$ for all natural number $k\geq1$.

For other kinds of model,we can similarly say that it is still possible:
$$\mathrm{RTIME(n^k)=co-RTIME(n^k)=ZPTIME_{\geq\frac12}(n)}$$ for all $k\geq1$.

Also, $\mathrm{UP=co-UP}$, etc.

And more importantly, all of these could SIMULTANEOUSLY hold if $\mathrm{L=CH}$ or $\mathrm{P=PP=CH}$ without violating any known complexity result.