CH3OH can act as a weak base. It does not do so by giving up an OH- group. That would leave a CH3+ behind, which is extremely unfavorable. When CH3OH acts as a base, it accepts a proton, becoming CH3OH2+.
CH3OH can also act as a weak acid. It does this by giving up a proton, becoming CH3O-.
In many respects, CH3OH is acting like water here. It is both a weak acid and a weak base, and has approximately equal strength in both of these roles. Overall, you might then consider it neutral. Depending on the context, however, you may need to consider it's ability to act as an acid and/or act as a base.
