Heelan was born in Andover, Massachusetts, the daughter of Actress and Lowell Public Schools Teacher Kimball Heelan and Playwright Kevin Heelan. She has a brother named Conor. Heelan's father, as well as being a dramatist, has been a faculty member at Phillips Andover Academy since 1983. Her parents met when her father was getting his Master's degree at Smith College.
Briga began dating singer Martin Johnson in 2009. They both attended Andover High School.
On Heelan's role in Ground Floor, TV critic Alan Sepinwall said Heelan is "simply terrific as Jenny: warm and quirky while always feeling like a strong and independent character". The Los Angeles Times said Heelan "makes Ground Floor watchable", describing her as "remarkably alive and in the moment; she makes real all that she touches. I could easily recommend tuning in just to watch her work – and hereby do."
On October 16, 2014, Heelan became engaged to her Ground Floor co-star Rene Gube. They were married on May 8, 2015. On October 31, 2016, it was announced they were expecting their first child, a daughter, due in March 2017. Heelan gave birth to their daughter Bennet Alejandra Gube on March 23, 2017.