"They stuff people's heads down the toilet the first day at Stonewall," he told Harry.
"Want to come upstairs and practice?"
"No, thanks," said Harry. "The poor toilet's never had anything as horrible as your head down it –– it might be sick." Then he ran, before Dudley could work out what he'd said.
(Harry Potter and the Sorcerer's Stone)
I am not sure if it refers to the toilet or has some other role in the sentence. What is its role?
Yes, it refers to the toilet. Harry has joked that the toilet might be ill if Harry stuffs Dudley's head down the toilet, because Dudley's head is more horrible than anything else anyone's ever put in the toilet. (Yikes!)
There's one other thing going on here: Dudley was intimidating Harry by suggesting that he would dunk Harry's head in the toilet, and Harry turned it around on him, suggesting that Harry would dunk Dudley's head in the toilet. This plays on the ambiguity in Dudley's statement.
Then, with the following sentence:
Then he ran, before Dudley could work out what he'd said.
Harry appears to be quick-witted and clever, while Dudley appears to be dull and slow-witted.
