Paramount said that it had stopped filming seven times during the pandemic, for reasons including UK government travel restrictions.
It alleges the Federal Insurance Company paid out only $5m (£3.6m), even though losses were many times that.
Federal parent company Chubb was approached for comment.
Mission: Impossible, which stars Tom Cruise, is a blockbuster franchise for Paramount.
The series of action movies has made hundreds of millions of dollars for the film studio. One movie alone, 2018's Mission: Impossible - Fallout, took more than $791m worldwide in box office takings.
But along with many other sectors, the film industry has been hit hard by the coronavirus crisis, with many cinemas closed for long periods around the world.
Film and TV productions have been disrupted, causing expenditure to soar, with the cost of testing, consultants and protective equipment adding millions of dollars to budgets.
Mr Cruise, who is also a producer on the film, apparently threatened to fire crew members after a breach on the set of Mission: Impossible 7 in England in December if they did not take Covid protocols seriously.