Let $f:X\to Y$ be a log resolution of normal varieties, let $\mathrm{Exc}(f)$ be the locus (the complement in $X$ of the largest open where $f$ is an isomorphism).
Is there an example of $f$, such that some codimension $1$ component of $\mathrm{Exc}(f)$ does not appear in the support of $K_X-f^*K_Y$?
Strictly speaking I don't think it makes sense to talk about the support of $K_X-f^*K_Y$. Both $K_X$ and $K_Y$ (and hence the expression you write) are linear equivalence classes of divisors, rather than actual divisors, so their support is not well-defined.
In spite of that, I think I get the motivation of your question; if so the answer is "yes". Let $f$ be the minimal resolution of a du Val singularity (a.k.a. Kleinian singularity, rational double point). Then $\operatorname{Exc}(f)$ is a collection of rational curves, in particular has at least 1 codimension 1 component, but $K_X=f^*K_Y$.