The assumed answer of this question has been yes, but one filmmaker, set to make a film about the history of the song, is questioning this assumption in a very public way. Jennifer Nelson filed for a class action suit against the publishing company, Warner/Chappell, who owns the right to the song's tune, originally a song called "Good Morning to All" written by two sisters in the late 19th Century.
Nelson's tentatively titled documentary "Happy Birthday" includes a scene where the song is performed, and she arranged for a $1,500 licensing fee to secure the rights to use the song. Eventually, though, she came to realize that, under her interpretation of the law, Warner/Chappell didn't own the rights to the tune. Richard Brauneis, a George Washington University law professor who agrees with her, and he's written a 68-page article making that claim.
The question is a particularly important one for documentary filmmakers who often want to mark the passage of another year in a subject's life and often use birthday parties to show this. On one hand, the filmmakers find it hard to make a fair use argument for use of the song, but on the other, they find the scenes in which it is included important ones to tell the story of individuals.
As the New York Times reports, Steve James paid $5,000 to include the song in his landmark documentary "Hoop Dreams."
According to the CNN Money report on the lawsuit, the filmmaker is claiming that the Warner/Chappell group only owns the rights to a more recent 1920s piano arrangement of the song.
Nelson's suit is seeking class action status, which would allow this lawsuit to fight to have all the previous royalties returned by Warner/Chappell. It is estimated by Nelson's lawyer (who spoke to the New York Times) that Warner/Chappell receives about $2 million a year in deals for the song.
The music lawyer Marc D. Ostrow has a fairly comprehensive and easy-to-read explanation of the legal status of performing, recording, and exhibiting the song on his website, available here.