Does a Beholder eye ray break its Invisibility?

[Dalebert]

You are saying here that they require a character to make a save or something detrimental will happen to them. That would be an attack action at my table. In this instance the Beholder would lose invisibility.

Yes, "at your table". Several of us who said that it's not an attack by RAW have agreed that many DMs would houserule that it becomes visible. No one is saying it's a wrong or bad ruling. When someone asks a question about the RAW, we give them an answer by the RAW.

But it's important to make a distinction between what's an attack or not. "Doing something bad" to someone is not necessarily an attack and the mechanical distinction is there for good reason. Mirror Image doesn't protect you from save-based spells or effects, for instance; only attacks. It's not intended to be that powerful.