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Well, i do not know the answer in general but since you are asking for a reference and if

there are some conditions on $A$ and $B$ that guarantee these are the only non-trivial ideals of $A \otimes B$?

There are some related results, for example:
Let $B$ an arbitrary algebra over a field.

Every ideal of the algebra $A\otimes B$, where $A$ is a central simple algebra, is of the form $A\otimes I$, where $I$ is an ideal of the algebra $B$.

This is theorem 4.3.2, p. 74, from the book of Drozd-Kirichenko.

Maybe this reference might also be of some interest to you.