The correct is "I do nothing but miss you."
The phrase "nothing but" is acting as an adverb. If this adverb is removed we get
I do miss you
This is correct. After "do" we use a bare infinitive.
The phrase "nothing but" is idiomatic and means "only". This whole expression means the same as
I do only miss you.
In the idiomatic expression, the structure is fixed. You to have "do nothing but". So ?"I nothing but miss her" is rather unusual and unidiomatic. This idiom requires "do support".
My first thought was that this was a parallel structure: "I do nothing, but I do miss you. But that is incorrect. For example, "He does nothing but miss her" is correct English, and if this was a parallel structure, it would be "... but misses her"