Greater Manchester Police (GMP) said the "joke" call came on 21 February after the fast food chain closed some UK outlets due delivery problems.
The caller said he "had an emergency" and had "to go to Burger King now".
A GMP spokesman said the man put lives at risk by keeping call handlers from answering real emergencies.
He said the call was made at 17:20 GMT and the force had received 74 emergency calls between 17:00 and 18:00 "of which 73 were answered.
At the time, handlers and officers were helping people trapped in a multiple vehicle crash, making a welfare check after a disturbance in someone's home, looking into a large group of youths drinking and causing a disturbance and dealing with the aftermath of an armed robbery.
During the call, which took 48 seconds, the call handler told him it was not a police matter and he should not contact police about it.
She also said he was "taking the mick" and she would not be "taking any sort of report from you".
Supt Mark Kenny said while some people will think it was "a harmless prank or joke, I want to reiterate the severity of this".
"We had 74 calls to 999 and 154 to 101 during the hour this came in, all of which needed our full attention," he said.
"Call handlers answered all but one of the emergency calls, but 21 non-emergencies went unanswered.
"People like this... put real strain on our already stretched teams - had this man not selfishly called for such a ridiculous reason, we would have been able to help more people."
Following the distribution problems, GMP urged people not to call the police over the fried chicken "crisis".