It's rare for a team to excuse a coach who has taken them to the Stanley Cup championship. Let's look at some recent times where it's happened:
Randy Carlyle - Anaheim Ducks
Cup Year: 2007
Reg season record after Cup: 182-135-35
Playoff record after Cup: 11-17
Lowdown: The Ducks could never re-capture their magic under Carlyle after their Cup, winning only one playoff series and finishing 2nd, 2nd, 4th, 2nd and 5th in their division upon his termination early in the 2011-12 season.
Did it work for team?: We'll vote yes, they hired Bruce Boudreau and are back near the top of the league and enjoying playoff runs again
Old coach's next gig: Later that year was hired by the Toronto Maple Leafs. He just got a contract extension, so it must be going great.
Peter Laviolette - Carolina Hurricanes
Cup Year: 2006
Reg season record after Cup: 95-78-16
Playoff record after Cup: 0-0
Lowdown: Injuries seemed to plague Carolina and they never got back to the playoffs under Laviolette, who was fired in December 2008 after another mediocre start. For whatever reason, the mercurial Canes just couldn't stay near the top.
Did it work out for the team?: Initially, yes. Paul Maurice came back, gave Carolina a boost and they made it to the Eastern Conference finals in 2009....But they haven't been back to the playoffs since.
Old coach's Next gig: Not until over a full year later, with the Philadelphia Flyers
John Tortorella - Tampa Bay Lightning
Cup Year: 2004
Reg season record after Cup: 118-108-20
Playoff record after Cup: 3-8
Lowdown: Torts seemed to change his style after the lockout, what once was a man who said "safe is death" reverted into a shot-blocking, defense collapsing dynamo. It didn't work out in Tampa, where he lasted 3 more seasons but failed to win another playoff series after winning the Cup.
Did it work for the team?: Negative. Tampa decided to hire Barry Melrose and it was an abject failure, they wouldn't make the playoffs for the next two seasons.
Old coach's Next gig: Not until mid-season of the following season, when he was hired by the Rangers and helped them into the 2009 playoffs, where they'd lose in the first round.
So those are the few cases in the past decade where a Cup winning coach was dismissed. What about those who have won but haven't gotten canned, what's been their secret?
Joel Quenneville - Chicago Blackhawks
Cup Years: 2010, 2013
Reg season record after (first) Cup: 171-83-40
Playoff record after Cup: 29-19
Lowdown: Got bounced in the first round the first two years after his Cup, but won it all in the third year and is a solid contender for a 3rd Cup this year.
DarrylSutter - Los Angeles Kings
Cup Year: 2012
Reg season record after Cup: 73-44-13
Playoff record after Cup: 15-15
Lowdown: The Kings got to Round 3 last year (and bounced by the Blackhawks) in their title defense, and now they are in 2-3 hole to Anaheim. They haven't had problems getting in and to the playoffs, but haven't been as convincing as their charmed 16-4 record in 2012.
Claude Julien - Boston Bruins
Cup Year: 2011
Reg season record after Cup: 127-62-19
Playoff record after Cup: 24-16
Lowdown: Lost in the first round of their title defense in 2012, but went to the Cup finals last year and are in a Game 7 to go back to the Eastern Conference finals this year.
Mike Babcock - Detroit Red Wings
Cup Year: 2008
Reg season record after Cup: 253-142-63
Playoff record after Cup: 36-36
Lowdown: Detroit has been on the downswing since their Cup- most their core of players has gracefully aged out and retired, and Babcock hasn't won but one playoff round in the past 3 years, and hasn't won more than 1 round since their trip to the Cup finals in '09. Still, despite that, Babcock has been seen as a source of strength and steadiness in a strong and steady organization.
And of course:
Dan Bylsma - Pittsburgh Penguins
Cup Year: 2009
Reg season record after Cup: 234-114-28
Playoff record after Cup: 27-27
Lowdown: It's been discussed before, but Bylsma is just 3-3 in playoff series in the past three seasons. He's had success in turbulent regular seasons, but has been unable to win a game in the conference finals since 2009.
Every coach in this sample that has not won a third round playoff game (except Bylsma) has been fired. But, to be fair, every coach that has been fired hasn't been out of the second round, and Bylsma has done that multiple times.
Any way you slice the data, it's not rocket science: the coaches that haven't succeeded, have been fired. Those that have continued to taste success with deep playoff runs, have kept their jobs.
If you think Bylsma has done enough with his very strong regular seasons, then you can likely point to him having a decent position among coaches post-Cup wins. If you already think the team needs a new direction, there's plenty of evidence above the show that Bylsma hasn't flourished like other coaches post-Cup win.