The London School of Economics and its students' union said the British broadcaster knowingly put LSE students' lives in danger and sullied the name of the LSE.
But the BBC disagreed and refused to withdraw the resulting documentary television program on its flagship investigative journalism series "Panorama," a report by the BBC said.
"Panorama" reporter John Sweeney posed as an LSE professor traveling with his wife and a cameraman on a study trip to film undercover for eight days.
The row comes amid increasing tension in the region over North Korea's anger at military drills by South Korea and the United States and Pyongyang's threat to test fire a missile close to the border of the two Koreas -- still officially at war since a 1953 cease-fire.
North Korea recently warned foreigners in South Korea that they could be in danger if hostilities broke out.
The LSE Students' Union's General Secretary Alex Peters-Day said the BBC used students "as a human shield."
But Sweeney said the students were told a journalist was with them but that the LSE wasn't because the trip wasn't an LSE excursion.
"All of them were told twice that a journalist was coming," Sweeney said.
Sweeney described North Korea as a "Nazi state" that practiced the "most extreme form of censorship," the BBC said.
He told a BBC Radio 4 news program that it was "extraordinary" for the LSE "to put words into the mouths" of the LSE students.
"What the LSE is saying we dispute. I can't talk for those students -- they are grown-up, they are brave and good people. All of those students could have dobbed me in. They didn't, the majority of these students support this program," Sweeney said.
Peters-Day, who wasn't on the trip, told the BBC News channel, "One of the students made it absolutely clear that she was not made aware of what happened."
"For us, this is a matter of student welfare," Peters-Day said. "Students were lied to, they weren't able to give their consent" and all LSE's future research was "now at risk."
The university said in an email sent to students and staff that the university's view was "students weren't given enough information" by the BBC.
It said the LSE was "fully supportive of the principle of investigative journalism in the public interest" but couldn't condone the use of (the LSE) name or the use of its students "as cover for such activities."
North Korea's government-run news agency KCNA reported Kim and senior military officers presented floral baskets in the name of North Korean workers party's Central Military Commission and National Defense Commission.
Japan and South Korea remained on high alert for any imminent ballistic missile test from North Korea, which observers said could be undertaken to coincide with the birthday observance.
North Korea's Committee for Peaceful Reunification of Korea rejected as insincere South Korean President Park Geun-hye's offer last week for a dialogue, KCNA reported.