Cineworld has pushed back the reopening of its UK and US cinemas to 31 July, blaming a delay in new film releases.
The world's second-largest cinema chain had previously announced cinemas in both countries would open on 10 July.
Cineworld said films including Mulan and Tenet had been delayed from July to August.
The company said cinemas would open with new safety measures in place, but some staff say the measures do not go far enough.
A petition signed by almost 1,500 people and started by Cineworld staff has called for masks to be made mandatory for customers in cinema foyers.
A leader of the Cineworld Action Group, who spoke to the BBC but did not want to give their name, said: "Some studies have shown that masks do help to protect. We don't have a choice about returning to work. If it makes it safer for us and for other customers, I think it's a small ask.
"It's a luxury to come to the cinema but our lives shouldn't be treated as a luxury, it's important."
The BBC has asked Cineworld to comment on the staff demands.
The cinema chain says new safety measures will include contactless payments and seating capacity limits in line with local regulations.
Advice for all cinemas from the film trade body, the UK Cinema Association, advises cinema companies to sell a reduced number of tickets to each screening, with people from different households sitting apart.
The association also says pick 'n' mix stands should be removed, and moviegoers should be encouraged to pre-order popcorn.
Earlier this year, the coronavirus pandemic forced Cineworld to close its 787 cinemas in 10 countries, while company bosses waived their salaries and bonuses.
The UK-based firm also halted dividend payments to mitigate the impact on trading.
Cineworld chief executive Mooky Greidinger said other films that will be released this summer include A Quiet Place Part II, The Broken Hearts Gallery, Bill & Ted Face the Music and Antebellum, as well as a re-release of Christopher Nolan's Inception on Imax.