So if you have a pretty bad hangover at the start of your run, you're already starting your run dehydrated, which is never a good thing. You might be able to get through a short run with a hangover (it still won't feel great), but it's dangerous to try to push through a long training run with a hangover because you could end up severely dehydrated.
You're also probably going to feel uncoordinated and clumsy when you have a hangover, which increases your risk for falling while running.
The bottom line: Try to save the alcoholic beverages for the night after your run. If you really want to have one alcoholic drink the night before a long run, make sure you're drinking plenty of water, too.