Wales Under-21 international Burns has scored one goal in 17 appearances for the Robins this season, but has started in only three games.
The 21-year-old has previously had loan spells with Forest Green Rovers, Oxford United and Cheltenham Town.
Burns could make his Fleetwood debut in the League One game against Scunthorpe United on Saturday.
Find all the latest football transfers on our dedicated page.

William Blake: Apprentice and Master has opened at the Ashmolean Museum after 19th Century floor plans were unearthed.
Blake created the majority of his books from a printing studio in the Hercules Buildings in Hercules Road, Lambeth.
The studio was destroyed when the building was razed in 1918.
Philip Pullman CBE, president of the Blake Society, said: "William Blake was a complete original; his power was his tenderness, his wit, his graphic line are like no-one else's.
"It's good to remind people every so often about this colossal imagination and his moral vision, which are just as potent now after two hundred years as they were when he brought them into the world."
The exhibition is on until 1 March 2015.

Grace Taylor, who was backpacking, had been uncontactable since 16 February, her mother Sam Taylor said on Facebook.
The family filed a missing person report to Dorset Police on 21 February.
Then on 22 February, Ms Taylor contacted her family, who booked her a flight home, although her mother said she did not board the flight.
Sam Taylor said her daughter had called from Ao Nang in Krabi province.
Mrs Taylor said Grace Taylor was "very stressed and frightened saying that people were trying to hurt her and were following her and she wanted to come home".
She added that "all previous backpacker sightings have reported that she is disorientated and not in a good mental state".
She appealed to the backpacker community to help locate her daughter.
A member of the family is flying to Thailand to search for Ms Taylor and bring her home, she said.

Rowett was furious after his side lost a two-goal lead to be held 2-2 at St Andrew's to end any hope of making the Championship play-offs.
"We've held on to leads at lot better this season," Morrison told BBC WM.
"We're probably far too open - we need to get that bit between our teeth."
Rowett called his side "unprofessional" and "lacking desire" after seeing Clayton Donaldson's brace cancelled out by two goals in the last 17 minutes, with Greg Cunningham's equaliser coming a minute from time.
"We need to see the game out, especially at home - they had no right to get back into it," said 28-year-old Morrison.
"It was very disappointing and a few choice words were said.
"Early on in the season, we had a tighter shape and people were working harder maybe to be more difficult to beat."
Blues went into the game with only a mathematical chance of finishing in the top six but the result has now extinguished their promotion hopes.
"It's a difficult time after such a good season and it turning out the way it has," added Morrison.
"It is an anti-climax, the fans are disappointed we're not still chasing the play-offs and that does affect the players but we're still going to put 100% in."

The 28-year-old was an ever-present in the side which won the County Championship title in 2016.
"He is one of the finest cricketers in the country," managing director of cricket Angus Fraser said.
"As a glove-man there are few better than him and he is capable of producing match-changing innings with the bat."
Since making his debut for Middlesex in June 2009, Simpson has made almost 250 appearances in all formats.
The left-hander has scored 6,687 runs for the club and has 472 dismissals to his name.
League One side Fleetwood Town have signed Bristol City striker Wes Burns on a youth loan deal until 8 May.
[NEXT_CONCEPT]
A recreation of the studio where William Blake created some of his most well-known work has opened in Oxford.
[NEXT_CONCEPT]
A 21-year-old woman from Swanage, Dorset is missing in south Thailand, according to her family, who have posted an online appeal for help.
[NEXT_CONCEPT]
Captain Michael Morrison says the Birmingham City players have accepted the criticism aimed at them by manager Gary Rowett after their draw with Preston North End.
[NEXT_CONCEPT]
Middlesex wicketkeeper-batsman John Simpson has extended his contract, keeping him at Lord's until the end of the 2019 season.