McGuire could be out until April after suffering knee damage in last week’s Super League round one defeat by Warrington Wolves.
Burrow, who took over after McGuire went off, will lead the team out as captain for the first time at Widnes Vikings on Sunday and retain the role for the following weekend’s World Club Challenge.
Burrow made his Rhinos debut in 2001 and has played 441 games for the club, including seven Grand Final victories.
Speaking to Rhinos’ club website, Burrow said: “I am absolutely honoured.
“Obviously I am just standing in for Danny, but to be asked I am really thankful.
“I love this club, I will captain them and I hope I will do them proud.”
With Beau Falloon sidelined by a broken toe, Burrow started at hooker against Warrington before moving to scrum-half when McGuire was injured.
He is confident being captain will not affect his natural game.
“I won’t have to do much more than I do generally,” he said.
“Organising teams is generally what hookers and half-backs do, so it will be pretty much that role anyway.
“Danny is still the captain and he is still here off the pitch, but maybe just on the pitch I will make a few decisions.”
He added: “The team is pretty well organised anyway by themselves
“There are a lot of leaders in the team and a lot of communication out there.
“It will be pretty easy and I’ll probably do very little, but if it needs me to make a decision, that’s the role.”
Sunday will see Burrow become Rhinos’ third captain in as many matches.
Kevin Sinfield, who held the role for 13 seasons, left the club after last year’s Grand Final triumph and McGuire was injured in his first game as permanent skipper.