If $f$ is rational, then your information will allow you to recover $f$, up to holes. That's an incredibly intense restriction on types of function though. I think Andre's answer makes it clear that there won't be many classes of functions that are less restrictive than this and meet your conditions.
To be complete: if $f$ and $g$ are known to be rational and have the same values on $\mathbb{Z}$, then $f-g$ is rational and equal to $0$ on $\mathbb{Z}$. Therefore the reduced numerator of $f-g$ is zero on $\mathbb{Z}$. A nonzero polynomial cannot have infinitely many roots. So the reduced numerator of $f-g$ is the zero polynomial. That means the unreduced numerator of $f-g$ is the zero polynomial, potentially with some holes at noninteger $x$-values. So $f-g=0$ except at finitely many $x$ where $f-g$ is undefined. At these $x$-values, either $f$ or $g$ is undefined. So $f=g$ up to where their nonintegral holes are.