European football's governing body UEFA - for whom Mr Taylor worked with distinction as general secretary and, more recently, executive director - confirmed the sad news after he had taken ill.
Lawwell said: "This is devastating news and our thoughts and prayers are with David's wife, Cathy, and their children, James and Alan.
"David was a fine man, a personal friend of mine and someone who made a great contribution to the game of football, both in Scotland and across Europe. He will be very sadly missed by everyone who knew him.
"On behalf of everyone at Celtic Football Club, I would like to extend our deepest condolences to David's family and friends at this very sad time."
A Scotland supporter since childhood, David became the first chief executive of the SFA in 1999, succeeding the long-serving secretary, Jim Farry.
SFA president Campbell Ogilvie said: "On behalf of his friends and former colleagues at the Scottish FA, and throughout Scottish football in general, I would to express our profound sadness at the tragic news of David's death.
"I would also wish to send our deepest condolences to his wife Cathy, and their children James and Alan, at this dreadful time.
"David was a solicitor to trade but it was his love of Scottish football and, in particular, the Scotland national team that stirred his passion during his tenure as chief executive.
"He was great company, had an astute business mind and a forward-thinking approach to the game that, allied to his patriotism as a member of the Scotland Supporters' Club, helped take the Scottish FA forward administratively.
"It was no surprise to any of us who witnessed his work at close hand when UEFA came calling for his services as general secretary in 2007. He has remained a friend and supporter of the Scottish FA and the game in general in this country and he will be dearly missed by all of us."
Born in Forfar in 1954, David was educated at Dundee High School before graduating from Edinburgh University with an LLB degree in law. He qualified as a Solicitor and practised until 1985, during which time he added to his legal qualifications with an MSc in Economics, and an MBA.
He joined the Scottish Development Agency (now Scottish Enterprise) in 1985, and held a succession of senior positions before being appointed the first Director of Scottish Trade International - a body set up to promote Scottish business overseas - prior to his appointment to the Scottish FA.