Generally, I don't use water from my asphalt roof for drinking (humans or animals) or anything I'm going to eat. There is a lot of nasty stuff in asphalt roofing. I use it for stuff like watering ornamental plants, washing the car, etc. Our chicken coop's roof is made from clear corrugated plastic roof panels, which are safer to drink from, so we do plan to use that once we get more rain barrels hooked up out there. And if we ever replace the roof on the house, we're getting a metal roof so we can use the water!
If you think your roof is old enough that the water might be safe, you can have the run-off analyzed by your local water testing laboratory. If you decide to have this water sample analyzed, have it analyzed for contaminants such as zinc, lead, chromium, arsenic, polyaromatic hydrocarbons, fecal coliform, and E. coli. Call your local extension for help finding the local water lab nearest you. It's usually not terribly expensive to have a water sample analyzed, so for me the peace of mind would be worth it.