Yes, if you apply for the Royalty membership you get a free birthday burger on your birthday. Also you will receive other perks, such as surprise and delight, and for every 9 burgers you get one free.
I am a rewards member and my birthday is 10/16. I did not get a free burger. Store said they no longer did that promotion, just the buy 9 and get 10th for free. I did eat there and choose email receipt from the table kiosk. The emailed receipt had a link to sign up to get a free birthday burger so I clicked on it and the link is dead. So IMO it is a little confusing as Red Robin has removed all mentions of the birthday burger. Bottomline: no birthday burger. Server did not do anything for my birthday but she also didn't know about it. She was covering half the floor already so we weren't about to take any of her time with sillyness.
on top of the birthdays burger rewards they might sing to you and give you a scoop of ice cream
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