What does the dog represent in the essay "A HANGING" BY GEORGE ORWELL?

I guess I would say that there are two ways you can look at this.

First, you can look at the dog as comic relief.  Comic relief is used in plays and such -- there's a bit of comedy during a tense time to let the audience relax a bit.  You could say that this is what is going on.

The other thing you can say is that the dog sort of indicates how inhumane the hanging is.  When the hanging is first starting, it works sort of like a machine.  But then when the dog comes in, all the participants become human again, in a sense.  They have to react to something in human ways instead of just carrying our their assignments like machines.

Since the dog makes them more human, it sort of emphasizes how inhumane the hanging itself is.