Gloucestershire said they were "deeply saddened" by the passing of "a club icon - known for his thunderous run-up, fearless fast bowling, and ever-present energy".
They described him as "a pioneer, a mentor, and a symbol of resilience" and added that "his contributions will never be forgotten".
Lawrence took 477 wickets at 31.27 in 170 first-class matches for the county, with a best of seven for 47 against Warwickshire.
"Firstly, everyone at the club sends their heartfelt condolences to Syd's family who supported him so well during his illness. As a club, we will support them in any way we can," said Gloucestershire club chair Peter Matthews.
"Syd was truly a Gloucestershire great, both on and off the field. As a player he was one of our greatest ever fast bowlers, who deservedly earned Test Caps whilst with us.
"On retirement, he continued to be heavily involved with the club, and was both a supportive and active president. He will be hugely missed."
Gloucestershire chief executive Neil Priscott added: "He was passionate about the power of cricket and how our sport can touch people's lives.
"Syd pushed us to reach out to communities far and wide and we shall continue to do that in his honour."
The England and Wales Cricket Board (ECB) also paid tribute to a "pioneering fast bowler whose career and character left an indelible mark on English cricket" and called Lawrence's 5-106 against the West Indies at the Oval in 1991 as his "finest moment" for his country.
"His impact on the game extended far beyond the boundary ropes," said ECB chair Richard Thompson.
"As a fast bowler, he thrilled crowds with his pace and passion. As a leader and advocate, he broke barriers and inspired change, becoming a powerful voice for inclusion and representation in our sport.
"Even in the face of his illness, David showed extraordinary strength and dignity, continuing to uplift others with his resilience and spirit. He leaves behind a legacy that will endure in the hearts of all who love cricket."