The 2-0 loss at Turf Moor left the Clarets two points away from the play-off positions, although they do have a game in hand.
"I am very saddened to receive the news and disappointed that I will no longer have the opportunity to achieve my ambition of reaching the Championship play-offs this season and, ultimately, promotion back to the Premier League," said Laws.
"I would like to take this opportunity to wish the club, the management, the players and the supporters all the very best for the future. I am taking advice from the LMA who will be working to reach an amicable agreement with the football club on my behalf."
Many supporters had called for Laws, 49, to be sacked during Tuesday's game, feeling that the club should be challenging for automatic promotion.
Barry Kilby, the Clarets chairman, told Laws on Wednesday that he was being told to leave.
"I have telephoned Brian today and have asked him to step down as manager of Burnley Football Club," said Kilby.
"Everyone is naturally disappointed and frustrated that recent results have not gone as we would have hoped."
It is not yet clear if Russ Wilcox, Laws' assistant at Turf Moor who he had first worked with at Glanford Park, has also been sacked. Burnley said they would make a further statement on Thursday.
Laws and Wilcox took charge of the Lancashire club last season following Owen Coyle's defection to Bolton Wanderers, but were unable to halt the slide towards relegation from the Premier League. In total they oversaw 44 games, registering 13 wins.