The ex-Sunderland and Ipswich Town boss, speaking to Sky Sports on Sunday, said that the Clarets paid the price for sacking the division's longest-serving manager with just eight games remaining.
The Irishman, a room-mate of Dyche's at Nottingham Forest, described the sacking as "crazy" after Burnley's six-year run in the top flight was brought to an end following a 2-1 defeat against Newcastle United at Turf Moor.
The outcome paved the way for Leeds United to stave off the drop back down to the Championship, despite starting the day in the bottom three by virtue of a far inferior goal difference.
Keane said: “Who said it was a good decision to get rid of Sean Dyche? Crazy. Sean Dyche would have done a good job if given the opportunity until the end of the season.
“He’s been here and done it, him and his staff. When he left, we said, oh there’s been a reaction, a reset. Rubbish. They’ve paid a heavy price for it.
“They should have stuck with Sean Dyche. Proven manager, great quality, done a brilliant job. No sympathy for Burnley.”