Let $A$ be the algebra $K[x,y]/(x^2,y^2,xy,yx)$. Then $A$ is a 3-dimensional commutative algebra and $Ext^i(M,A) \neq 0$ for any indecomposable non-projective module $M$ for some $i>0$. Namely: $Ext^2(M,A)=Ext^1(\Omega^1(M),A) \neq 0$, since $\Omega^1(M)$ is a direct sum of simple modules.

Now do we have the property

$Ext_B^i(M,B) \neq 0$ for any indecomposable non-projective module $M$ for some $i>0$ (*)

also for the enveloping algebra $B=A^{op} \otimes_K A$ ?

This is a 9-dimensional commutative algebra with Loewy length 3. The same trick does not work since $\Omega^1(M)$ is just a Loewy length at most 2 module. But maybe one can classify the $B$-modules that are first (or higher) syzygy modules and show (*)?