Hello, Look at these explanations I have found in a dictionary: 1. For the sake of somebody/something (also for sb’s/sth’s sake): in order to help, improve, or please someone or something: ● He moved to the seaside for the sake of his health. ● I only went for Kay’s sake. ● I hope he’s told the truth for his own sake (=because it will be good for him). 2. For the sake of it: if you do something for the sake of it, you do it because you want to and not for any particular reason: ● She likes spending money just for the sake of it.

"Sake" is a funny word, because it doesn't translate well and it's difficult for native English speakers to explain (it's just a natural phrase to us). Depending on the context, you could understand it as "think of what might happen to your eyes" or "do it for the benefit of your eyes".

I've never heard that saying used in the U.S.

i think its a romantic dialogue...

Likewise I haven't heard of it. Ar a guess, however, I'd use it in a context such as: Stop looking directly at the sun, for the sake of your eyes (literal). Probably meant in a metaphorical way, e.g something one shouldn't do, as if someone were to see it, their eyes would be damaged or equivalent. All speculation though :)