In the second episode of season one, titled Mole Hunt, Sterling Archer wakes up after a one night stand and discovers that his new lady friend has brought her pet pug, Abelard, into his apartment.
He is furious at his beleaguered and long-suffering butler, Woodhouse, for allowing him, in his drunken state the previous evening, to violate his own policy of not permitting dogs in the apartment1.
Much later, in episode 10 of the fourth season, titled Un Chien Tangerine, we learn that Archer doesn't actually dislike dogs - in fact, he immediately takes to the title character2 of the episode, an enormous English Mastiff named Kazak.
He explains that he loves dogs, with the notable exception of the dog his mother owned when he was a child, an unbelievably pampered Afghan Hound named Duchess.
Archer's animosity towards Duchess has nothing to do with her being a dog; it was actually motivated by his mother's deliberate and blatant favoritism towards the dog and corresponding emotional neglect of Archer. In fact, throughout the series, Archer has consistently shown himself to be incredibly fond of animals of all kinds. In particular, he was extremely excited to see someone's privately owned pet tiger, and he has horsed around with Cheryl's pet ocelot, Babu, on several occasions.
So if Archer doesn't hate any dogs aside from Duchess, why doesn't he allow dogs in his apartment?
1In Woodhouse's defense, he did try to stop Archer from bringing the pug inside, but Archer drunkenly refused to listen, insisting that the dog was barking the song Putting On The Ritz (it wasn't). Unsurprisingly, this didn't prevent Archer from threatening to rub coarse grade sand into Woodhouse's "tiny dead eyes" the next morning.
2The word Chien is French for "dog", and the episode is set in Tangiers, where Archer and Lana find Kazak; hence, Kazak is Un Chien Tangerine, or A Dog From Tangiers.