Are there any side effects to drinking alcohol

Violent solutions to problems and a high body count increase the level of Chaos in the city.

When the city is a violent chaotic place, people react accordingly. There are more guards and more traps. There are more rats to take advantage of the abundance of food. Essentially, if you signal to the game that a fight is what you're after, then a fight is what you'll get. The final mission, in particular, is dramatically altered by the level of chaos you've engendered or avoided up to that point.

Additionally, Dishonored has several endings, and chaos is the primary determinant of which of these you'll see.

In general, the optional missions that you should not complete are fairly obvious.

Specifically, you are correct, and the Granny Rags mission does increase Chaos. The Outsider is not specifically determined to keep it at a minimum, so his endorsement does not help in this regard. The Bottle Gang is helping to keep the plague at a minimum by selling their cheaper version of the Elixir; indeed, in a later mission you find a note that seems to indicate that even the guards use it if something happens to their ration. After you taint their stash, you spread the rat plague until they figure out what's going on, and the source of cheap Elixir is gone.