He led a selection committee that picked the Olympic team after months of work, and his role during the 12-day tournament would have been largely behind the scenes in a supporting role.
"To hear that he had been hurt and to know that he's not going to be coming over is a huge disappointment," U.S. coach Dan Bylsma said.
Shero, who is general manager of the Pittsburgh Penguins, will be acting GM in Sochi.
"Having worked with Dave for eight years in Nashville, and knowing, as I told the team, how much passion, time and effort he has put into putting this team together, it's disappointing," Shero said.
Poile won the Lester Patrick award in 2001 for his contributions to hockey in the U.S., and is a three-time finalist for NHL GM of the year. He has also filled various roles with USA Hockey.
He was the associate GM for the U.S. at the Olympics four years ago, when he assisted Burke after he dealt with tragedy. Burke's 21-year-old son, Brendan, died Feb. 5, 2010, in a car accident, and he skipped the opening ceremony of the Vancouver Games.
AP Sports Writer Greg Beacham contributed to this report.