Actually, you answered your own question.

You said it was true in the normal case, well that's not entirely true.

Independence is equivalent to the covariance being zero if the joint distribution is normal.

And you do get a zero covariance if you have equal variances of X and Y....

Cov(X+Y,X-Y)=V(X)-V(Y).

So let X and Y be indep Normals with different variances.