Edit: ok, for some weird reasons, i thought S and T have to be subrings of a unitary ring! lol the problem is just asking for any rings, which makes things much
easier and Opalg's example is perfectly fine! my apologies Opalg!
If you want unitary subrings, how about looking at subrings of the 3×3 matrices over the integers? You could take S to be all those of the form , and T to be all those of the form . Then S∪T is not closed under addition.
(I'm sure there are simpler examples that a bona fide ring theorist could provide.)