On a day where Australia's Test top three fell cheaply against India in Adelaide, Burns (74) and Labuschagne (58 not out) gave a reminder of their credentials.
Burns was bowled by Riley Meredith in the last over of day three on a Bellerive Oval deck that has flattened since 18 wickets fell on day one.
The pair shared a 126-run stand as the Bulls reached 2-164 at stumps in pursuit of a mammoth 414.
Queensland still need another 250 runs to win but Labuschagne said hopes were high in the dressing room.
"Joe batted so well and we were really enjoying it out there. It was devastating to lose him at the end," Labuschagne said.
"We've batted really well on this ground before. There's a lot of confidence in our shed and if we can do the hard work tomorrow morning it could be a good finish tomorrow afternoon."
Tasmania earlier posted 353 on Sunday, with veteran George Bailey reaching his 24th first-class century.
The veteran hit 13 fours and a six in his knock of 109 that comes after he stepped down as Shield captain last month to focus on his batting.
He was out edging to first slip off Mark Steketee, but not before helping bat the Tigers into a commanding position.
Bulls' legspinner Mitchell Swepson (4-99) cleaned up the Tigers' tail after lunch, including a stunning one-handed leaping caught-and-bowled to remove Alex Pyecroft (29).
Queensland lost out-of-sorts opener Matthew Renshaw early as they got to tea at 1-49.
An aggressive Renshaw hit two leg-side sixes before being bowled by ex-Test quick Jackson Bird for 27 off 24 balls.
It was a golden opportunity missed for Renshaw, as questions linger over whether Aaron Finch has the temperament to remain as opener for Australia's Test side.
Resuming the day at 4-240, Tasmania lost Matthew Wade (40) who failed to add to his overnight score.
Wade, the leading Shield run scorer this season, top scored with a valuable 63 in Tasmania's first innings of 167 after they were sent in.
Queensland were routed for 107 reply, with Pyecroft taking 5-18 in his second first-class match.