I think the subcategory of symmetric bimodules is never closed under extensions for non-semisimple local Artin algebras (that are not necessarily commutative).

Namely, assume that this subcategory is closed under extensions. It contains the unique simple bimodule and thus every finitely generated bimodule since it is closed under extensions. Thus one just has to find one finitely generated bimodule that is not symmetric. In case the algebra is non-commutative, one can take the regular module.

In case the algebra is commutative, can one always take the Jacobson radical of the enveloping algebra or someone has a better suggestion?

Here a general abstract argument that shows that the subcategory of symmetric bimodules is never extension closed in case $A$ is a commutative finite dimensional Frobenius algebra that is not semisimple:

Let $A$ be such an algebra and $B=A \otimes_K A$ its enveloping algebra and assume that the subcategory of symmetric bimodules is closed under extensions. The simple module $S$ is symmetric and thus the subcategory of symmetric finite dimensional bimodules equals the module category of $B$. It thus also contains $B$, but $Hom_B(A,B) \cong D(A) \cong A$ has dimension less than $B$ and thus $B$ is never symmetric. This is a contradiction and thus the subcategory of symmetric bimodules is never closed under extensions.