Let $H$ be a minor of $G$. By letting $G$ be an even cycle, we see that $\chi(H) \leq \chi(G)$ does not hold. Is there some other upper bound that would hold, for instance is $\chi(H) \leq \chi(G) + c$ true, where $c$ is some constant?
I really only care about edge contractions, but if a good bound can be obtained for a minor (i.e., when allowing for edge and vertex deletion too) that's also fine.
The question Can contracting an edge increase the chromatic number by more than one? says that if we just contract a single edge, we indeed have that $c = 1$. But what if an arbitrary sequence of contractions is allowed?
You cannot get a bound of the form $\chi(G) + c$ (although I'm sure other, weaker bounds do exist).
There are lots of ways to see this, but probably the easiest is by the following construction. Let $H$ be the complete graph $K_n$. Make a graph $G$ from $H$ by replacing each edge with a path of length $2$. This new graph $G$ is bipartite, and hence has $\chi(G) = 2$ (to see this, make all the old $K_n$ vertices red, and all the new vertices in the middle of the `edges' blue).
Now we have that $H$ is a minor of $G$ (just contract all the paths of length $2$ back into edges), but $\chi(H) - \chi(G) = n-2$, which we can make as large as we like by picking a bigger $n$.
Of course, if your sequence of contractions has only $k$ contractions, then $\chi(H) \leq \chi(G) + k$.