The asylum-seekers, from Sri Lanka and the Philippines, housed Mr Snowden for two weeks when he fled the US after leaking thousands of files.
A government spokesman said that there were no substantial grounds for believing that the claimants would face danger in their home countries.
The group, who are also seeking asylum in Canada, have 14 days to appeal.
The asylum-seekers - a Sri Lankan couple with two children, a Sri Lankan man and a Filipino mother with a daughter - were introduced to Edward Snowden by their lawyer, Robert Tibbo.
He brought him to their homes in 2013 just after the former National Security Agency contractor revealed he had leaked classified information to the press revealing widespread US government surveillance.
The identity of the group, who sheltered him for about two weeks, only became public shortly before the release of an Oliver Stone film about the whistleblower.
"We now have less than two weeks to submit appeals before the families are deported," Mr Tibbo said, calling the decision "completely unreasonable".
There are currently 8,956 asylum seekers in Hong Kong awaiting a decision on their case, according to the Immigration Department.
But the territory only recognised 72 people as refugees between 2009 and December 2016, the South China Morning Post reported in February.
Most spend years in limbo waiting for the government - which is not a signatory to the UN refugee convention, but considers claims for protection on the grounds that those sent home could be tortured - to process their case.