Burrow, who was 41, had lived with motor neurone disease (MND) since being diagnosed in late 2019.
The diagnosis came just two years after he retired from playing, following a stellar 17-year career that included winning eight Super League Grand Finals, three World Club Challenges and two Challenge Cups.
Burrow's death was announced by Leeds Rhinos, who called their former scrum-half "a true inspiration throughout his life whether that was on the rugby league field or during his battle with MND".
In 2022, Burrow was presented with the Helen Rollason Award at the BBC's Sports Personality of the Year ceremony, with his ex-Leeds team-mate Kevin Sinfield also recognised.
Burrow was given the award for relentless fundraising and raising awareness of MND while battling the illness himself.
Burrow spent his entire club career with Leeds and made more than 400 appearances between 2001 and 2017.
One of the most successful rugby league players in history, Burrow was made an MBE in the 2021 New Year Honours for his services to the sport and for his work in the MND community.
Burrow and his great friend Sinfield were both made CBEs in the 2024 New Year Honours. The pair have helped to raise more than £15m for MND charities since Burrow's diagnosis.
Burrow struck a defiant tone on learning he had MND.
"The worst thing for me is people pitying me," he said. "I know it's going to come, but I want to be as normal as ever.
"While I am able-bodied and feel fit and strong and healthy, I want to do normal things and not be treated any differently."
In announcing his death on Sunday, Leeds said of Burrow: "He never allowed others to define what he could achieve and believed in his own ability to do more.
"The outpouring of love and support that Rob and the whole Burrow family have received over the last four and a half years meant so much to Rob.
"In particular, the rugby league family and MND community have rallied around Rob to inspire him, thank you for your support.
"For those who knew Rob throughout his life, his determination and spirit in the face of MND over the last four and a half years came as no surprise.
"Rob never accepted that he couldn't do something. He just found his way of doing it better than anyone else.
"He will continue to inspire us all every day. In a world full of adversity. We must dare to dream."