Per the New Jersey Herald News and the Te Deum Blogspot, Fr. Louis Scurti, a campus minister at William Paterson University in New Jersey, "brings his two dogs everywhere [oh?] and that includes Sunday Mass." His pair of pooches set themselves up in the sanctuary during Mass, "making people feel included" [huh?] and providing a "symbol of domesticity" [double huh?]. Although the apparently untethered canines "have been known to growl" at late-comers, Fr. Scurti assures us that his dogs "don't remove the sacredness of the liturgy at all."
The dictates of common sense are hard to put into words. If one has to explain to a pastor why his mutts don't belong in Mass, one goes into the effort with the uneasy feeling that such words might be wasted on, well, someone who needs that kind of thing explained in the first place. But most folks can tell the difference between a liturgy and a living room, and many Catholics are out of patience with priests (granted, in shrinking numbers) who still treat the Mass as their personal property.
Yes, I could cite some canons: 1983 CIC 1210 excludes from churches things that do not advance "worship, piety, or religion"; Canon 1220 states that "whatever is inappropriate to the holiness of the place is excluded"; and Canon 285 tells priests to avoid "unbecoming" behavior at all times; but none of these norms quite make the point I want.
I might try appeal to the Catechism of the Catholic Church, paragraph 1181, which observes that a church "ought to be in good taste and a worthy place for prayer and sacred ceremonial" and that "the harmony of the signs that make it up should show Christ to be present and active in this place", but that doesn't quite get me there either.
So, I guess we need a new law: No animals in the sanctuary, ever, and no animals (except certified assistance dogs) in a church. There, now I won't be tempted to bring variously my two dogs, four parakeets, a cockatiel, salamander, newt, corn snake, or rabbit to church, and my fellow worshippers will just have to seek elsewhere for "the sense of calm and peace" I experience around my pets. Though I still think observing common sense would save us all a lot of trouble.
Oh, one last thought: it sometimes happens that, if canon law does not adequately address a problem, civil law might. The appropriate university or diocesan officials might want to check New Jersey leash laws. Liability issues are always better explored before problems arise, rather than after.
Update Sept 13: Several people have offered other examples of priests who bring dogs to Mass with them, whether in or out of the sanctuary. To them I say, first, you are not alone in thinking such practices are quite out of place in public worship, and second, every pastor has a boss. See also Jimmy Akin, Curt Jester, BettNet, and more from Te Deum.
Update Nov 9, 2007: A priest has been removed from ministry for, among other things, permitting dogs to roam in the sanctuary during Mass.