Why? Lol. It's a once-in-a-life-time event. In short, no. That's the right of the monarch to be coronated at whatever tax-payer expense.
I'm not in England so I don't know what the economic concerns are regarding the cost of the coronation, but I don't think it would be appropriate for the monarch to have to take additional cost of the coronation--that's not to say some of the after party, but the coronation itself, no!
Now, I could see making the monarch pay for the birthday celebrations if that's not done already. Birthdays are egotistical and all the pagan kings kept their birthdays, so I could understand forgoing the Trooping of the Colour or making them pay for it if they want to throw a shin-dig for themselves.