What happens if someone with mutations to alcohol dehydrogenase (ADH) drinks alcohol?
People, which have no ADH wouldn't be able to process methanol. Since the specificity of the ADH is higher for ethanol you get quite large amounts of ethanol for days (until the methanol level is under a specific level). Another possibility is to treat patient with the ADH inhibitor Fomepizole.

This also makes problems for the ethanol metabolization, but according to this paper ("The biochemistry of alcohol metabolism - a brief review") there are two other pathways of breakdown. One takes place in the peroxisomes and utilizes catalase and hydrogen peroxide (but is limited by the availability of peroxide), the other take place in the smooth endoplasmatic reticulum and uses the Cytochrome P450 enzyme.

I am not sure, if complete losses of ADH really happen (haven't found any reference) but mutations in this enzyme are quite common leading to a lower enzyme activity.