Canonist Ed Peters writes:
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.
Actually, I think that one of the canons Ed mentions–1220–would prohibit what the priest is doing. The canon reads:
Can. 1220 ยง1. All those responsible are to take care that in churches such cleanliness and beauty are preserved as befit a house of God and that whatever is inappropriate to the holiness of the place is excluded.
The highlighted phrase is meant precisely as a catchall to cover all the myriad "Please don’t eat the dasies" situations that the legislator knows he can’t individually envision. Having dogs in the sanctuary–at least dogs who are not assistance animals–is inappropriate to the holiness of the place, and thus they are to be excluded.
By including the highlighted phrase, the legislator intends "those responsible" (including the priest) to exercise common sense in figuring out what is inappropriate to the holiness of the place and–if the priest in question has defective common sense on this point–then someone higher up (his bishop, the CDW) should explain it to him.
After all, dogs are irrational creatures that cannot appreciate the sacred and therefore do not belong in sacred places.
Our Lord may have been making a point about humans when he said it, but the foundation of the metaphor in the animal kingdom makes the same point with even more force:
"Do not give dogs what is holy; and do not throw your pearls before swine, lest they trample them under foot" (Matt. 7:6).

