Burns sustained a dislocated collarbone after falling awkwardly during the first half of his side's 4-1 victory.
Burns may be out of action for four to five weeks and is set to miss the 7 February County Antrim Shield Final.
Mitchell was helped off in the second half with an ankle injury and faces a two-month spell on the sidelines.
Striker Mitchell, who is the league's leading scorer with 19 goals, will visit a specialist this week to have the injury assessed.
The Dungannon forward last week agreed a pre-contract deal to play for Glenavon next season and will join the Mourneview Park club in the summer.
Linfield manager David Healy lamented the loss of Burns for an important period of the season.
"Aaron is a big player for us and has scored important goals. He could be out for at least four to five weeks," said the Blues' boss.
Linfield midfielder Aaron Burns and Dungannon Swifts striker Andy Mitchell suffered injury setbacks in Saturday's Premiership match at Stangmore Park.