Burney will enter his 21st season as an NFL assistant, most recently working for the Washington Redskins from 2010-14.
"Jacob is a great taskmaster with tremendous experience," said Lewis. "He started under Bill Belichick at Cleveland and has worked under some other great head coaches as well since he came into the league."
Lazor was fired as the Miami Dolphins' offensive coordinator on Nov. 30, and will replace Ken Zampese, who was promoted to offensive coordinator when Hue Jackson was hired as the Cleveland Browns' head coach.
"With Ken's elevation to coordinator, I'm ecstatic about bringing in Bill," Lewis said. "He's bright, and energetic and he also has experience under great head coaches. He's got everything we're looking for to continue to uplift our offense."
Coyle spent 2001-11 on the Bengals' staff before leaving to become the Dolphins' defensive coordinator. He was also fired during the regular season.
"After being away for four years as a coordinator, Kevin comes back as another asset for (Bengals coordinator) Paul Guenther," Lewis said. "We know Kevin as a great coach of secondary technique and as a coach with great ability to add ideas for game planning."