a euphemism for performing oral sex, usually cunnilingus. The allusion is to both eating out and to shape of receiver's body and spread legs forming the shape of the letter Y. Sometimes abbreviated as DATY.

when you're hungover as shit after a spending a night drunk as a skunk

Humorous way to allude to the kind of menu you will be having for tonite's dinner. You will probably choose some tuna taco , or perhaps bearded clam or maybe loose meat sandwich or even some snapper.
For those who need further explanation, the diner is called the Y not because its in some gym, but due to the peculiar shape of all aforementioned dishes.

To have a wonderful romantic dinner eating at the YMCA.

A euphemism for performing oral sex, usually cunnilingus, as others have mentioned. "The Y" is also a nickname for the YMCA, once a famed hangout for trolling gay men. Also, only men have a Y chromosome, so dining at the Y could also mean fallatio, or perhaps analingus as well.