The thieves melted away with the pictures, which showed scenes of the film's two main characters the snowman and the young boy James.
Film bosses believe the cells were stolen to order by or for someone with extensive knowledge of animation.
John Coates, the film's producer, said: "These cells were some of the few remaining as most have been sold or perished since the film was made.
"They were due to be exhibited in Japan and Australia later this year and are irreplaceable.
"I am very saddened that these pieces will be lost to generations of people who love the film. These pictures have enormous personal and sentimental value to me."
They have been placed on a register of stolen art and police carried out forensic tests at the building, he said.
The Snowman, an adaptation of Raymond Briggs's classic wordless children's book, has been broadcast in over 100 countries.
It has won numerous awards, including a BAFTA and an Oscar nomination, and has become one of Britain's Christmas television favourites.