I have not seen anything to indicate airport authorities or airfield owners will be responsible for collecting the tax. This is obviously the way visitors are expected to pay, but ensuring they do is another matter:
An alternative to payment by (form) F24 is forseen for those persons unable to pay with this model: payment may also be made through a bank in Euros in favor of Bilancio dello Stato al Capo 8 – Capitolo 1223. The transfer must refer to BIC: BITAITRRENT, the reason for transfer: data of the person liable to pay tax, identification of the aircraft (nationality and registration marks), the tax code and the reference period and finally the number IBAN - IT35 Z010 0003 2453 4800 8122 300.