There are also examples where the minimal Martin boundary is a proper set and is closed when the random walk is not symmetric. I don't know if you can find any such example with a symmetric one.

In a slightly different context, Hueber and Müller showed that the Martin boundary for a continuous random walk on the Heisenberg group $H^3(\mathbb{R})$, the Martin boundary is a disc, while the minimal Martin boundary is a circle (see "asymptotics for some Green kernels onthe Heisenberg group and the Martin boundary”. In:Mathematische Annalen283 (1989), pp. 97–119.)

Actually, if $\Gamma$ is a nilpotent group with some nice condition of generation (for instance finitely generated), positive harmonic functions are constant on the cosets of the commutator subgroup. That was proved by Margulis (see "positive harmonic functions on nilpotent groups”. In:Dok-lady Akademii nauk SSSR166 (1966). In Russian. English translation SovietMathematics Doklady, 7, 1966, 241-244, pp. 1054–1057.)

Thus, a positive harmonic function $f$ defines a positive harmonic function on the abelian group $\Gamma/[\Gamma,\Gamma]$. You can easily show that minimal harmonic functions on $\Gamma$ define minimal harmonic functions on the abelianized group, so that the minimal Martin boundary reduce to the minimal Martin boundary of an abelian group.

Now, if the random walk is non-centered on an abelian group, its minimal boundary is a sphere, so it is compact. This gives other examples taking nilpotent groups with a non-minimal Martin boundary (as the Heisenberg group, according to Hueber and Müller).
Though, to my knowledge, few is known in general for nilpotent groups.