England face an uphill battle in the second Test in Adelaide after closing on 17 for two in response to Australia's 473 for nine declared, Burns the first to go after edging Starc into the slip cordon in the third over.
The Surrey opener has already fallen twice to Starc and Cook fears the match-up will become increasingly one-sided unless his fellow left-hander alters something.
It has been a forgettable tour for Burns, who went down in Ashes folklore for all the wrong reasons after being bowled from the first ball of the series, and he has contributed just 17 runs from three innings so far.
His idiosyncratic batting style has come under the microscope after a mixed bag this year, with one ton and three fifties but also six ducks in 10 Tests, and Cook believes Burns needs to do something to become more reliable.
"You're always told in international cricket 'do what you do in county cricket, you don't change how you play, you'd be mad to change it straightaway'," said Cook on BT Sport.
"It's probably getting to a stage now where he'll have to sit down and go 'this is what I've got with my technique, if I want to play even longer for England and play 60, 70, 80 Test matches for England, what do I need to improve?'
"This does work for Rory Burns on occasion but to be more consistent early on in his innings, I think he needs to really think about how he gets that bat playing coming down straighter.
"It's in Rory Burns' hands, still. He's definitely going to bat again unless something else extraordinary happens in this Test match. If he scores a hundred in the second innings he then gets to play some more games."
Cook, England's all-time leading Test run-scorer and one of the finest openers the country has produced, feels Burns gets his hands too far away from his chest, which leaves him out of position against a bowler like Starc.
"The perfect Mitchell Starc angle, to hit that ball perfectly with it coming away and his bat coming across, your timing has to be absolutely spot on," added Cook, who acknowledged there are perils of making technical tweaks mid-series.
"If it's not spot on, the face of your bat is slightly closed as you come to hit it. It's a technique he now has to decide what he's going to do because Mitchell Starc for the next five Test matches if he stays fit, it's almost the perfect combination against him.
"The problem is tinkering with your technique in the middle of a massive series, you're then batting in a game maybe thinking about technique is not what you want to be doing.
"The only thing you want to be doing is committing or watching the ball as it's coming down. You've got to keep your mind as clear as you can."