I think no inversion (DO first and S-V then) is needed in this case as there is no DO... I would say "The more forests disappear" is right (my teachers used to tell me: "the more you read, the more you learn", which is a similar construction, "the more"+S+V), however, I've found on the Internet some examples like this: "The more trees that die, the more trees that will die." (
[link]:yyeqjGVherkJ:dieoff.org/page47.htm+%22the+more+trees%22&hl=es).
Swan explains in his grammar (p. 123, Practical English Usage, OUP) that the word order in both clauses in this kind of structures is this one: "the"+comparative expression+subject+verb. He also notes that sometimes "that" is used before the first verb, as in my last example above.
Hope this helps!