Burns made scores of 8, 51*, 0 and 4 in the opening two matches of the series to extend his lean first-class season to 125 runs at 10.41.
After his unbeaten half-century in Australia's chase in Adelaide, he was caught behind off Jasprit Bumrah in the first innings in Melbourne and then off Umesh Yadav in the second.
"Joe Burns has been released from the squad and will return to the Brisbane Heat," national selector Trevor Hohns said. "Unfortunately, Joe's returns haven't been what he or the selectors would like or what we consider he is capable of."
Speaking before the fourth day's play in Melbourne, head coach Justin Langer, who has been a big supporter of Burns, hinted time had run out. "Runs are the currency of value, performance is the currency of value, you have to keep performing to be selected," he said.
The omission of Burns means that Australia will have a new opening combination in the third Test. Pucovski is shaping for a potential debut at the SCG after recovering from the concussion he suffered in the first warm-up match against India earlier this month.
Warner is on the comeback from the groin injury he sustained in the second ODI although it is not yet confirmed that he will be available. He netted extensively at the MCG while the second Test was ongoing, but Langer has previously flagged uncertainty over his availability.
Warner and Sean Abbott, who is also included in the 18-man squad after recovering from a calf strain, had to train separately from the Test group in Melbourne having been in Sydney when the current Covid-19 outbreak started. They will rejoin the biosecure bubble on Thursday.
"David has made strong progress in his recovery from injury and will be given every chance to play in Sydney with another seven days until the match," Hohns said. "Sean has fully recovered from a calf strain and is also available for selection.
"Will is in the final stages of the graduated return to play protocols and has been symptom free for some time. He will be fit to play in Sydney subject to completing the return to play protocols and an independent assessment."
Squad: Tim Paine (capt), Sean Abbott, Pat Cummins, Cameron Green, Josh Hazlewood, Marcus Harris, Travis Head, Moises Henriques, Marnus Labuschagne, Nathan Lyon, Michael Neser, James Pattinson, Will Pucovski, Steve Smith, Mitchell Starc, Mitchell Swepson, Matthew Wade, David Warner
Andrew McGlashan is a deputy editor at ESPNcricinfo