But the museum defended the show, saying it showcased ways that design can help tackle problems including environmental issues.
"We are surprised that the campaign took the decision to withdraw from the shortlist," the Design Museum told Dezeen.
"The Beazley Designs of the Year exhibition embodies the museum's purpose to make the impact of design visible and highlights the role that design can play in providing solutions to many of the challenges we face today - including challenges to the environment."
"No awards on a dead planet"
Beazley Designs of the Year is an annual awards programme recognising the best designs from around the world over the past 12 months. Nominated projects are exhibited at the Design Museum each September, with winners announced by the end of the year.
Projects that seek to tackle environmental or social issues have featured heavily on recent shortlists. Last year's overall winner was Forensic Architecture, a group that uses architectural techniques to investigate human-rights abuses, while Ikea's flat-pack refugee shelter won in 2016.
Beazley Designs of the Year curator Gemma Curtin told Dezeen in 2017 that the programme showed how designers were increasingly "looking to create cleaner water, purer air, less pollution, less waste. All of these issues seem to be key."
The first winner of the award, then called Brit Insurance Designs of the Year, was the Plumen low-energy lightbulb in 2011.
Extinction Rebellion captured attention this year when its activists shut down key parts of London to protest government and industrial inaction in the face of climate change.
Its distinctive logo, posters and flags were designed collectively by their art group and made available for anyone to download and use to spread their message.
But Extinction Rebellion has refused to allow these designs to be included in the Design Museum's 2019 exhibition, saying there can be "no awards on a dead planet".
Extinction Rebellion accused museum of artwashing
The group objected to the show's sponsor, claiming Beazley "runs counter to our movement's values". The specialist insurer provides cover against political, environmental and cyber risks, among others.
"This is yet another example of a major cultural institution attempting to artwash the unacceptable behaviours of its financial backers," said Clare Farrell, part of Extinction Rebellion's art group.
Artwashing is a political term used by activists to condemn the presence of art or artists being used to disguise anti-social issues such as gentrification, or the practice of companies funding arts events to distract from their negative activities.
"For Extinction Rebellion to be co-opted by an organisation like Beazley runs counter to our movement's values," Farrell added.
"We are in the business of refusing business as usual, and the insurance industry, which supports the mitigation of financial loss caused by immoral environmental practices, is not something we can have anything to do with."
"We call upon other designers who have been invited to contribute to the show to follow our lead and also tell the truth by refusing inclusion in the exhibition.”
"Awards have made a positive contribution to the world"
The Design Museum defended the awards show and its sponsor against the boycott.
"Visited by thousands of visitors and learning groups every year, the awards have made a positive contribution to the world we live in, demonstrating over the last 12 years that designers are working on valuable projects that improve lives," it said in a statement.
"The Design Museum is grateful for the continued support from Beazley in making this exhibition possible.”
Extinction Rebellion is using the boycott to demand that the insurance industry acknowledge the risks of climate change.
"The insurance industry is the one part of our economy that is designed to have regard to the longer term future," said Tim Crosland, part of Extinction Rebellion's legal strategy team.
"We call upon the Insurance Industry to tell the truth to the British Public, which is that without urgent and radical action on the climate and wider ecological crisis their insurance policies are at risk," Crosland added.
"Unless and until they do the right thing we cannot accept this award nomination.”
Following weeks of action by Extinction Rebellion earlier this year the UK's parliament declared a climate emergency.
A series of reports from the UN report have warned that there are just 12 years left to avert disastrous global temperature rise above 1.5 degrees, and that a million species of plants and animals are at risk of extinction due to human activity.
Last year several designers demanded that their work be removed from an exhibition at the Design Museum because it was hired as the venue for an arms-industry event.