All grape products, including and especially wine, are toxic to canines. Even a small amount is not fit to be eaten by a dog. The halakha is that things which are not fit to be consumed by a dog do not need a kosher lepesah hekhsher. Why then does wine, which is not fit to be consumed by a dog, need a kosher lepesah hekhsher?
During the Exodus from Egypt, it says that "Lo Yecheratz kelev leshono." This cannot refer to barking, as barking is performed with the mouth and throat, not the tongue. Rather, this is stating that the dogs in Egypt refrained from drinking things that are sharp on the tongue i.e.: alcohol.
In order to reward them for this tee-totaling behavior, Hashem gave them the basar treifah, since "ayn simcha elah bibasar viyayin" and the dogs abstained from wine. This was important because "mitzvah gedolah lihiyos bisimcha tamid," and Hashem wanted to ensure that the dogs were not inadvertently being mevatel an aseh. Obviously this worked, since dogs are nearly always happy.
To return to the original question, canine inability to imbibe alcohol stems from their choice to be Prohibitionists and not due to physiological phenomena. But since it is theoretically possible for them to drink, alcohol still requires certification for Pesach.
Due to generations of abstention (and a lack of thumbs), they're just REALLY BAD at holding their liquor.
It would be an insult to a person to rely on the psak of a dog (that it is not edible) when the person enjoys it. You are saying that the dog has better taste (pun intended) than the human being.
Besides, the fact that we use wine in the Bais Hamikdash means that the Torah has given the psak that no matter what the dog says, it is edible. Thus, it needs a hechsher.
There obviously is a different opinion regarding dogs and wine. See the Chicago Tribune quoting Dr. Jim Sokolowski - veterinary nutritionist that wine or beer in moderation won't hurt a pooch. My dog loves and drinks wine all the time. See here in Bishulei Gilyonei Rosh Hashana 4a questioning why can't a dog drink wine? למה לא ישתה כלב יין?