This is a dogs' instinct and is all about resources, by chasing away cats the dog will have access to more prey to sustain itself and its pack.
A pack of dogs will chase away other dogs too - for the same reason. It is the same with foxes - dogs will chase them.
Foxes will try to chase away cats too - even if they have limited chance of success against an adult cat.
And as you know, cats chase away other cats to avoid competition over the same resources.
If a dog tries to chase away a cat, but the cat refuses to move, the dog will often withdraw from a potential fight (take a look on youtube bad ass cats and you will understand what I mean).
Every predatory animal has this instinct to remove competition - both pack animals and the solitary ones; packs of lions chase away the young males and keep the females (they do most of the hunting), so it makes sense to have them in the pack.