Hennessey spent seven seasons at Palace but lost his position as the club's number one goalkeeper to Vicente Guaita in the last two campaigns.
The 34-year-old is Burnley's second summer signing after the arrival of Nathan Collins from Stoke City and will jostle for a spot between the sticks with Nick Pope.
"Popey's a great lad but it's competition, which is healthy," Hennessey told the club's website.
"That's part of the goalkeeping union. There's always competition. There's always that one spot but we'll be friends off the field.
"That's part of the goalkeeping union. There's always competition. There's always that one spot but we'll be friends off the field.
"There's a fantastic spirit here and hopefully we'll do really well this season.
"We've always known what Burnley are like. A great atmosphere. The lads are superb, always tight and have always played really well together and with Sean Dyche, as well, have an amazing manager.
"They're exciting times and I'm excited to start working with the lads."
Hennessey has won 96 caps for the Welsh national team but was an unused substitute in all their four matches at this summer's European Championship.
