Maximal subcoalgebras of an $F+1$-coalgebra corresponding to an $F$-coalgebra
Suppose that the functor $F$ preserves monos. In fact, I am going to assume something slightly stronger, namely that $S \subseteq T$ implies $F(S) \subseteq F(T)$ (otherwise we have to insert isomorphisms here and there). Your Kripke functors are of this kind, I think.

First a little diversion. Let $G$ be a functor from sets to sets which preserves the subset relation. Consider a coalgebra $s : S \to G(S)$ and a subset $A \subseteq S$. Call a subset $T \subseteq S$ "good" if the restriction $s_T : T \to G(S)$ of $s$ to $T$ gives a coalgebra $s_T : T \to G(T)$ and the image $s(T)$ is disjoint from $A$. Any union of good subsets is again a good subset. Indeed, suppose $T_i$'s are good and let $T = \bigcup_i T_i$. Then $s$ restricted to $T$ maps into $\bigcup_i G(T_i) \subseteq G(T)$, and obviously $T$ is disjoint from $A$. Therefore, there exists the largest good subset of $T$, namely the union of all the good ones.

Now apply the previous paragraph to the functor $G(X) = F(X) + 1$ and $A = \lbrace \mathrm{inr}()\rbrace$ to get the desired result.