Tax is regressive and unfair as it lays the greatest burden on the least well-off. Tax cannot either distinguish between abusive and responsible drinking patterns and is therefore ineffective in addressing alcohol misuse. Taxes have very little influence on alcohol misuse, yet high taxes carry unintended consequences, such as increased incentives for informal and unregulated markets to thrive.
The heaviest drinkers/alcohol abusers are least responsive to price changes. Therefore, the heaviest drinkers tend to continue consuming alcoholic beverages even if they are more expensive.
The less well-off heavy drinkers try to maintain their alcohol intake by either switching to cheaper drinks of the same category - or to a different category - or turn to the unregulated market (illicit drinks that are potentially unsafe).
On the other hand, moderate drinkers – which are the vast majority - are more responsive to price changes and therefore are likely to reduce their consumption after price increases, which, in turn, may lead to reduced health benefits.
Further reading and scientific reference here (IARD) and here (UNESDA).