  For a fixed finite family of graphs $\mathcal{F}$, the
$\mathcal{F}$-Minor-Free Deletion problem takes as input a graph $G$ and an
integer $\ell$ and asks whether there exists a set $X \subseteq V(G)$ of size
at most $\ell$ such that $G-X$ is $\mathcal{F}$-minor-free. For
$\mathcal{F}=\{K_2\}$ and $\mathcal{F}=\{K_3\}$ this encodes Vertex Cover and
Feedback Vertex Set respectively. When parameterized by the feedback vertex
number of $G$ these two problems are known to admit a polynomial kernelization.
Such a polynomial kernelization also exists for any $\mathcal{F}$ containing a
planar graph but no forests. In this paper we show that
$\mathcal{F}$-Minor-Free Deletion parameterized by the feedback vertex number
is MK[2]-hard for $\mathcal{F} = \{P_3\}$. This rules out the existence of a
polynomial kernel assuming $NP \subseteq coNP/poly$, and also gives evidence
that the problem does not admit a polynomial Turing kernel. Our hardness result
generalizes to any $\mathcal{F}$ not containing a $P_3$-subgraph-free graph,
using as parameter the vertex-deletion distance to treewidth
$mintw(\mathcal{F})$, where $mintw(\mathcal{F})$ denotes the minimum treewidth
of the graphs in $\mathcal{F}$. For the other case, where $\mathcal{F}$
contains a $P_3$-subgraph-free graph, we present a polynomial Turing
kernelization. Our results extend to $\mathcal{F}$-Subgraph-Free Deletion.
