The 22-year-old rejected the offer of a new contract from the Hammers to make the switch to Turf Moor, where he joins former Bournemouth frontman Danny Ings as a new recruit.
"I need a fresh start and I am looking forward to playing for Burnley. It's all about me playing football now," said Hines.
"Burnley seems like a good club to me so it wasn't really hard to decide to come here.
"I'm not a person who talks too much. I prefer to do it on the pitch, but I think I can do a lot in the Championship for Burnley."
Hines, a loan target for Clarets boss Eddie Howe last season, made 31 appearances for the Irons after coming through the youth system at Upton Park.