I can't imagine why you'd have an opening three or four inches below the top of the bath, with no drain tube. That would just cause the bath to overflow several minutes earlier. It would be better not to have such a hole, and save your bathroom floor for a few minutes longer.[...] You asked, however what the "opening" is called, which I'm not sure you really mean.
If you're only talking about a tub in which there is no drain tube, then I guess we'd call it an overflow opening or something - depending upon context.
No. Imagine a brand new bath in the showroom or warehouse; the overflow hole would indeed be "a hole" but, were you to refer to it, perhaps you would have to distinguish it from the hole that is the "plug hole" - it therefore becomes "the overflow hole."... but simply querying sdg's "a tub in which there is no drain tube". In fact, such a thing would just be a hole,
You have assumed that the bath is complete and in situ and functional. I do not know if that is what the OP intended. She may well be wishing to specify that hole and/or, as above, distinguish it from the plug hole.However, if the conversation is just about the visible parts of the bath, I suggest that the opening can be called just "the overflow"