Benning says the veteran forward is "going to be out a while" and that there is no timeline for his return.
"He's experiencing some of the symptoms of the COVID. I guess they call it being a long hauler," the GM said on Monday. "He still has some issues he's working through with that.
"So once he's feeling better, the first step for him is to get feeling better and then once he's healthy enough and feeling better, he'll start working out and get back out on the ice."
Benning added that Sutter does not have myocarditis, a condition that causes heart inflammation and has been associated with COVID-19.
Sportsnet's Iain MacIntyre says Sutter's fatigue only became debilitating by mid-August.
At least two NHL players have been sidelined indefinitely due to heart inflammation, including Edmonton Oilers backup goalie Alex Stalock and defenceman Josh Archibald.
Sutter, a veteran centre, was one 21 Canucks players and four coaches who tested positive during an outbreak in late March, postponing several games and keeping the team off the ice for more than three weeks.
Many experienced symptoms, including extreme fatigue, fever, chills and shortness of breath. Loved ones also fell ill, including wives and children.
Every player on the Canucks opening-night roster is fully vaccinated, Benning confirmed Monday.
Sutter signed a one-year, US$1.25-million contract with the Canucks in July.
The 32-year-old American has played six seasons in Vancouver but his tenure has been marred by various injuries. Last season, he put up 12 points (nine goals, three assists) in 43 games.
Sutter is one of several Canucks who are out as Vancouver prepares to kick off the regular season -- and a six-game road trip -- Wednesday against the Oilers in Edmonton.
Defenceman Travis Hamonic has yet to join the team due to what Benning has described as "personal reasons."
The Canucks put Hamonic on waivers Sunday and have assigned him to the American Hockey League's Abbotsford Canucks to free up space on the roster.
"It just buys us more time, we save money on the cap and we get a roster spot to get through the (upcoming six-game road) trip," Benning said.
"He's through working through some personal matters. We're going to loan him to Abbotsford and once he has the personal matters cleared up, he can come in, start skating with them and be ready to join the team when we get back."
Right-winger Brock Boeser missed the final five games of the pre-season with an undisclosed injury and Benning said he is considered "day-to-day."
Depth forward Tyler Motte is still rehabbing after undergoing off-season surgery for an upper-body injury and no timeline has been set for his comeback.