The incidents included 74 on ballistic missile submarines. Three happened while the vessels were in naval bases; one of these was on a ballistic missile submarine, the Ministry of Defence said.
Peter Luff, Minister for Defence Equipment, Support and Technology, gave the figures in response to a parliamentary question from Scottish National Party defence spokesman Angus Robertson.
Of the 266 fires, 243 were classed as "small-scale" and categorised as a localised fire, such as a minor electrical fault creating smoke, the BBC reported.
There were 20 medium-scale fires that were generally categorised as a localised fire, such as a failure of mechanical equipment creating smoke and flame, requiring the use of "significant onboard resources".
Reacting to the MoD report, Robertson said: "Any one of these fires could have had catastrophic consequences and the frequency of these incidents raises the most serious safety concerns.
"That so many of these incidents occurred on submarines that may have been nuclear armed is deeply troubling.
"It makes a mockery of any UK claims to having a credible "independent" nuclear deterrent."
Information on whether ballistic missile submarines were armed with nuclear weapons when the fires occurred was not available, Luff said.
Britain's nuclear submarines are based at Faslane on the Clyde but the location of the incidents was not specified.