Here's how the balloon feature usually works: When you sign up for Twitter, the site asks for your birthdate. When you view your own profile on that date, a note under your bio wishes you a "Happy Birthday" and a delightful little animation of multi-colored balloons floats across screen (PS tap to pop them). If you've made your birthdate settings public, the balloons appear to anyone visiting your profile, along with a message reading, "Today is their birthday!" Click the message, and the balloons appear again.
I did some digging, and I still see them on desktop on profiles that share their birthday publicly. But on the mobile app, the balloons are gone. And though "Today is their birthday!" still appears, you can't click on it for more balloons. Users are also reporting that they do not see the balloon animation when viewing their own profile on their birthday.