I am struggling with the following problem:
Being given two non-isomorphic intuitionistic Kripke frames $F_1$ and $F_2$ there is a formula $\phi$ such as: $$ F_1 \models \phi\ and\ F_2 \not \models \phi. $$ I would like to use this result to prove that logics (considered as set of formulas) of two non-isomorphic intuitionistic Kripke frames are different.
I agree with Noah in the comment. You cannot find $\phi$ in general, not even for finite frames. You can find trivial examples by yourself.
However, if $\phi$ exists, then your final claim follows immediately from the definition of the logic of an intuitionistic Kripke frame.
You can find $\phi$ if $F_i$ are rooted and finite. (I think there are multiple ways to prove this. I see it in the following way. If $F_i$ satisfy the same formulas, then $F_1$ is a generated subframe of a bounded morphic image (also called reduction) of a disjoint union of copies of $F_2$, and vice versa -- this is Birkhoff's theorem from universal algebra in action. This implies $|F_1|\leq|F_2|$ and $|F_2|\leq|F_1|$ and it becomes clear that $F_1\cong F_2$.)
Hope this helps.