If I'm going to be climbing a mountain, would drinking a small amount at altitude have any negative effects (besides general alcohol effects)?
The effects of alcohol are exacerbated at altitude, each ounce of alcoholic beverage will have a greater perceived effect on the consumer. This is in part due to the lower oxygen level in the blood of anyone at altitude, usually this is most notable at or above 8,000ft.
There are some ways to mitigate the effects of alcohol, and speed recovery.
Alcohol will cause several detrimental effects pertaining to climbing as well.
This doesn't mean you shouldn't drink any alcohol on a climbing trip, but you must be aware of it's effects and how to deal with them. All-in-all, use moderation and good judgement.
Although the quantity consumed and the altitude where consumption occurs may change this answer, as long as it is in moderation; I'm not aware of any specific problems.
I have known people who bring along a hip flask full whiskey. I could not participate as I was underage at the time, but I was told that it helped him get to sleep, and made sore bones/muscles hurt a little less at night.
Without knowing just how high you are going and how much alcohol you want to consume, it's difficult to say, however, to be safe I'd avoid any alcohol at altitude...
Beyond alcohol's well know effect of causing mental and physical impairment (even with small quantities of alcohol), the vasodilative (widening of veins and arteries) and diuretic (increasing urine excretion) effects of alcohol would only exacerbate symptoms of altitude sickness such as dehydration and oedema. Vasodilation will also cause you to loose body heat more quickly (despite alcohol's feeling of giving you warmth - in fact that's why it gives you that feeling).