Trump's long-standing territorial designs on Greenland, an autonomous Danish territory, triggered the most serious crisis in NATO's history since it was formed in 1949.
The mayor of Greenland's capital called on media professionals and content creators to act responsibly on Thursday after a failed attempt by a German comedian to hoist the US flag.
Bavarian comedian Maxi Schafroth, 41, attempted to run the Stars and Stripes up a flagpole near the cultural centre in Nuuk, before he was confronted by angry passers-by.
When questioned, he claimed to be a US official before leaving to disapproving looks from locals, according to an journalist for the AFP news agency who was at the scene.
He was reported to the police and fined, local media said.
Avaaraq Olsen, mayor of Kommuneqarfik Sermersooq district which includes the capital, was outraged.
"Raising a flag at our capital cultural centre, the flag of a military superpower that for weeks has been implying military force against our country, is not a joke," she said in a statement.
"It is not funny. It is immensely harmful."
Olsen said Greenlanders, particularly children, were worried and afraid about the current situation.
"When you amplify those fears for content, clicks or laughs, you are not being bold or creative," she said.
"You are adding to the distress of an already vulnerable population...So, pause before you film. Think before you stage something 'funny.' Consider whether your next question or stunt will inform the world or simply make a child cry or a family feel less safe in their own country."
Schafroth is known in Germany for his involvement in the satirical programme "Extra Drei," broadcast by the regional NDR channel and attracting more than one million viewers.
In a statement to Germany's Der Spiegel and the Ritzau agency, the channel expressed its "regrets" after the incident.
Greenland's government is working to alleviate the stress and anxiety on the population caused by US President Donald Trump's repeated threats to seize the Arctic territory.
Trump's long-standing territorial designs on Greenland, an autonomous Danish territory, have triggered the most serious crisis in NATO's history since it was formed in 1949.
The US president said owning Greenland was crucial for US national security and claimed, without providing evidence, that if the US didn't seize control of the territory, Russia or China would.
He has since backtracked on a threat of using force to acquire Greenland and begun talks with Copenhagen and Nuuk.