Let $\phi$ be a modal sentence. Let $M$ be a model and $w$ be a world. Is it true that either $w\models\phi$ or $w\models\neg \phi$?
I feel it's wrong, but what's wrong about this argument? If $w\models \phi$, then we are done. If not, then $w\not\models\phi$, i.e., $w\models \neg \phi$. Is the last step wrong? Is it true only for atomic sentences that $w\not\models p$ iff $w\models \neg p$?
What a couterexample to the claim could be?
Yes, the Kripke semantics of modal propositional logic tell us that at a given world, either $\phi$ or $\lnot \phi$ holds. The clause for negation in the usual inductive definition of satisfaction at a world is that $\lnot\phi$ holds at $w$ if and only if $\phi$ does not hold at $w.$ (More generally, all propositional connectives obey their usual truth tables.)
You might be thinking of Kripke semantics for intuitionistic propositional logic. Here, accessibility must be reflexive and transitive, and if a statement holds at $w$, it must hold at every accessible world. The negation clause is that $\lnot\phi$ holds at $w$ if and only if $\phi$ does not hold at any world accessible from $w$. This leaves open the possibility that $\phi$ does not hold at a world $w,$ but it holds at some $w'$ accessible from $w,$ so that $\lnot \phi$ does not hold at $w$ either.