Young people are being warned of the dangers of getting involved with the game, which is thought to have originated in Australia before making its way to the UK and Ireland, after two people died.
Those taking part are supposed to down, or 'neck', alcoholic drinks in one go, posting a video online of their efforts and then nominating others to try to 'go one better'.
Under pressure to take bigger and bigger risks, some videos have shown people drinking pints of WD40 or urine - or drinking and then carrying out a dangerous stunt.
The family of a 19-year-old man who drowned in Carlow on Saturday say he was taking part in the game at the time.
Real friends don't Neknominate. You have the right to say: 'No'. Join the thousands of others who are standing up against this 'game'.
Public Health Agency
Johnny Byrne died after jumping into the River Barrow at Milford bridge.
It is understood that his brother entered the water to try to save him, but had to be rescued. He later took to Facebook to hit out at the craze.
"My young 19-year-old brother died tonight in the middle of his nomination," Patrick Byrne posted.
"He thought he had to try and beat the competition - after he necked his pint, he jumped into the river. If people have any decency and respect, they will refrain from anymore of this stupid neknomination."
Earlier that same day, a young DJ was found dead by his friends in Dublin in another case believed to be linked to the drinking game.
Gardaí are awaiting the results of a post mortem to establish the exact cause of his death.
The Northern Ireland Facebook page, which is followed by nearly 11,000 people, has been coming under fire for promoting such a dangerous game.
On Sunday, the creators said they were no longer continuing to post the videos.
"Some people just can't keep things fun and have to take it over the line. End of the road for us as far as neck nomination videos go," they said.
They also posted a final video, featuring a glass of water instead of alcohol and calling for others to stop participating in the game.
But while the Irish Communications Minister Pat Rabbitte has called for Facebook to take direct action, the social media network has refused as it does not consider the videos to break its rules on harmful content.
We do not tolerate content which is directly harmful, for example bullying, but controversial or offensive behaviour is not necessarily against our rules.
DUP Upper Bann MP David Simpson has criticised Facebook for its stance.
"Facebook has stated that these videos do not breach their 'community standards'," he said.
"But this is a very questionable stance, given that one of these standards states it is against the promotion of self-harm. Whatever views someone may hold about this 'game' at its outset, there can be little doubt now that it has morphed into something which is now very dangerous with a pressure being exerted on people to consume ever more dangerous substances."
Mr Simpson added: "The neknomination craze relied upon peer pressure from friends, but it also shines a light on the wider pressure that young people can face online.
"Government cannot regulate everything which appears online but there must be a greater collaborative effort to ensure that people content can be reported and that people can find the help and support they need when problems do arise."
The Public Health Agency has called for people to recognise the peer pressure involved in such a game and to take a stand against it.
"Drinking 'games' are not new, but they have the potential to quickly increase consumption of alcohol and this can be highly dangerous," a spokeswoman warned.
The agency notes that drinking makes you more likely to take risks, but it impairs your judgement and makes you less aware of the dangers, while also making you less physically competent.
"Social media has increased the effect of peer pressure, and we need to try to harness this in a positive way - using the media to reinforce messages about the risks of drinking, and how important it is to look after your friends," the PHA spokeswoman added.
The Young Farmers Clubs of Ulster organisation is among those urging its members to be responsible and avoid the game.
"We urge our members to say no to the 'nek and nominate' game, and we offer our condolences to the families of the young men who are believed to have died while playing this game," YFCU President Martyn Blair said.
"Please think about the tragic consequences that may result from participating in this craze."
Peer pressure can be a very powerful thing, and people can get caught up in it.
Public Health Agency
But SDLP West Tyrone MLA Joe Byrne said that he had spotted new 'neknomination' videos appearing online in the last 24 hours and urged people to stop.
"This is not a game. It is destroying lives," he said.
"I would call on people to take these concerns seriously and look after their health and wellbeing by not participating in this 'game', and to protect their friends from the serious risks associated with it by not nominating them to take part."
Sinn Féin Chair of Omagh District Council Marty McColgan said the craze was exacerbating existing problems with alcohol and drug abuse.
"These deaths must act as a wakeup call," he added.
"I would urge people to be responsible, not partake in the game and, if nominated, break the link and not encourage what has become a dangerous and lethal online sensation."
Chair of the Independent Panel on the Responsible Retailing and Promotion of Alcohol, Duncan McCausland, added: "This is precisely why the work of the panel is so important. It is imperative that we take a zero-tolerance approach to harmful drinking practices.
"Whilst many of these dares do not take place on licensed premises, it is important for the licensed trade to make it clear that such dares will not be tolerated in their pubs and clubs."
Pubs of Ulster Chief Executive Colin Neill also condemned the 'neknomination' craze.
"Alcohol is a regulated substance for good reason and we strongly encourage people to drink responsibly and not to pressurise friends and peers into downing crazy amounts of alcohol," he said.