FAQ for
Escape from New York (1981)

It is unknown as to why he specifically breaks the tape, but one answer is, short of Castle or Carpenter themselves, is that he is just the stereotypical anti-hero, and would rather see the world remain in chaos instead of getting better, seeing as they used him to try and improve the state of the world.

A more logical reason is that he did it in retaliation as he felt cheated by Hauk because he was not told about the explosive charges in his neck until AFTER he agreed to take on the mission and the injection and thereby endangering his life. This is the same as his motive for doing a similar thing in Escape From L.A. (1996) when he shuts down the Earth.

A third reason could be that Snake broke the tape out of respect for those who helped him escape (Cabby, Maggie, Brain). Keep in mind, the President (Donald Pleasance) wasn't too appreciative of the others (saying they did a "service to their country") and was more worried about how he was going to look on TV to address the summit leaders.