The variable approaches zero.
This is correct.
The variable approximates to zero.
This is grammatically correct, but actually means the variable is almost zero (for example 0.0001). It doesn't convey any movement of the variable.
The variable are closing zero.
This is incorrect, but "The variable closes towards zero" is fine for casual conversation. For a situation where perfect grammar is important though, this is not recommended.
The variable becomes more and more equal to zero.
I guess this is kind of okay. A lot of native speakers would definitely say this, but it's really incorrect because equal is binary. You can't really be a little bit equal (even though lots of people would still use it that way).
As for conveying the opposite meaning, there's actually not many options. "The variable goes away from zero." sounds a bit awkward, but I can't think of a better way to word it off the top of my head. Maybe someone else here can think of a good phrasing. Personally, I'd try a different approach altogether ("The variable's value increases").