  Let $L$ be a distributive lattice and $R[L]$ the associated Hibi ring. We
show that if $L$ is planar, then any bounded Hibi subring of $R[L]$ has a
quadratic Gr\"obner basis. We characterize all planar distributive lattices $L$
for which any proper rank bounded Hibi subring of $R[L]$ has a linear
resolution. Moreover, if $R[L]$ is linearly related for a lattice $L$, we find
all the rank bounded Hibi subrings of $R[L]$ which are linearly related too.
