As a Catholic school survivor, this struck me as really, really funny. By the time I made it back to my desk, I had the whole scene in my head:
Doddery, nearly-blind old priest shakily holds up a wafer during a hot summer's Mass. What little breathable air that is left in the church is being gasped at by the parishioners as they wait in line for communion. The St. John's ambulance staff outside get ready for that Sunday's heat stroke victims.
A dog has trotted into the church through doors left open in the hopes of admitting a thus-far absent breeze. He butts in line in front of a twelve-year-old boy too shocked to see a dog in a church to do anything, and before anyone thinks to speak out stands on his hind legs in begging position, triggered by the sight of the wafer raised just the way his master lifts a Milk-Bone into the air at treat time.
Priest: The body of Christ.
The priest places the wafer on the dog's tongue and gets ready to present communion to the next person. It strikes him that the elderly man he just served had an awfully hairy face, but it takes all sorts to make a world. He decides it's nice to know some people still receive communion hands-free, the old way.
As it turns out, this is one time that the truth is less funny than the fiction. It wasn't a Catholic church at all, but an Anglican one. This makes it less funny. First of all, a lot of Anglican churches have a day where they bless their parishioners' pets once a year, so a dog in an Anglican church is less funny. Second, Eddie Izzard already covered most of the relevant topics about how the Church of England is funny in his "Cake or Death" monologue:
Although, if he'd had a dog get some cake, that would have been even funnier....