The word "Fire" is never uttered, for the reason you mention.
Theatres have a fire code word, such as "Mr Sands" so "Mr. Sands is in the bar" would mean that it was getting a bit warm where the drinks were served.
The fire is then assessed. If it is serious enough (In our theatre every fire is serious enough because we don't have a protected auditorium, but a proper theatre has masses of fireproof doors and walls) then the performance is stopped, and the audience is told there are technical difficulties and asked to follow the directions of staff and to wait outside the building.
In a proper theatre, the audience shouldn't know there's a fire until they are outside of the building and they see fire engines arriving. Also in theory a modern theatre should protect the entire audience from a raging inferno for nearly an hour.
PS
@Illaveago: if you are in a theatre and get told there's a technical problem and you need to leave, please don't now shout "They're lying to you: It's a fire..."