The prime minister’s office has also been accused of failing to understand Northern Ireland after a post on Instagram showed photographs of a showcase of the region's culture during an event attended by PM Rishi Sunak.
The initial post included an Irish flag emoticon. It is understood that the flag was included in error and was quickly deleted.
TUV vice-chairman Allister Kyle said: “While some unionists may dismiss this as an innocent oversight I would not be inclined to be so generous. Many families in Northern Ireland know what it cost to keep Northern Ireland as part of the UK as it is a gross insult to those who paid the ultimate sacrifice to display a foreign flag when referring to Northern Ireland.
“This government has repeatedly paid lip service to being a Conservative and Unionist government but their actions, whether it be on the Protocol or even their posting on social media, show how much they value Northern Ireland.”
Councillor Kyle added: “Unionists have received a reminder of where the Protocol will drive Northern Ireland on the government’s own social media account.”
Ulster Unionist leader Doug Beattie said the incident is the latest which shows that Westminster "doesn't understand" Northern Ireland.
"I'm not one of these people who get all wound up about flags or whether someone has put something on by mistake," he told the BBC.
"It doesn't make sense that they would do that – if they felt the need to put a flag up, it should have been the Union flag."
The mistake came after Northern Ireland was left out of the latest round of Levelling Up funding.
Mr Beattie said: "There are people out there who want to hand all of the devolved power we have back to Westminster, and here is what Westminster is doing - they got rid of our Levelling Up funding, they brought us the Northern Ireland Protocol, they brought us the Windsor Framework, and now they're firing out things on Instagram where they are using the wrong flag.
"These people don't understand us, they don't know us. We are better served with devolved government."