Edinburgh-born Weir, who earned 61 caps between 1990 and 2000, revealed his diagnosis in 2017 to raise awareness of the condition.
"Doddie was an inspirational force of nature," his family said in a statement posted on the Scottish Rugby website.
"His unending energy and drive, and his strength of character powered him through his rugby and business careers and, we believe, enabled him to fight the effects of MND for so many years.
"MND took so much from Doddie, but never his spirit and determination. He battled MND so bravely and, whilst his own battle may be over, his fight continues through his foundation until a cure is found for all those with this devastating disease."
In 2019 Weir was awarded an OBE for services to rugby, motor neurone disease research and the Scottish Borders community.
