If you set it to hide your date of birth, then I’m pretty sure it won’t appear in the birthday section. Indeed, unless your birthday is fairly soon, I think it will cease to be an issue. The forums are moving to Discourse soon, and I don’t think it has that feature (although I could be wrong there :)).
I’ve always been away on my birthday, so I don’t know if mine shows up - and we’ll have moved to Discourse before my birthday this year, so I’ll never know. :bawling: