The group said they would reschedule the gigs due to take place in Glasgow on Friday, and at the O2 in London on Monday, Tuesday and Wednesday.
They did not say who had tested positive for coronavirus.
The Last Domino? tour started in September after being postponed by almost a year because of the pandemic.
Frontman Phil Collins, 70, has been performing seated and has not been able to play the drums because of ongoing health issues, including back problems following surgery.
He told the BBC last month that he can now "barely hold a [drum] stick", and has been replaced behind the drum kit by his 20-year-old son Nicholas.
The pair have been joined on the road for the tour - the band's first since 2007 - by keyboardist Tony Banks and guitarist Mike Rutherford, both aged 71 and both founding members of the group.
Genesis had six number one albums in the UK in the 1980s and 1990s, with hits including Land of Confusion, Invisible Touch and I Can't Dance.
"This is a hugely frustrating development for the band who are devastated with this unlucky turn of events," a statement from the band said.
"They hate having to take these steps but the safety of the audience and touring crew has to take priority. They look forward to seeing you upon their return."
The band, who have sold more than 21 million albums in the US, are due to start the North American leg of the tour in Chicago on 15 November.