When royal forces defeated papal forces in 1870 and overthrew the Papal States, one of the various unsuccessful attempts to mollify the pope was a law saying that no postage need be paid on letters to the pope, sent from anywhere in Italy. Does that law still exist?
The Vatican has its own postal operator (Vatican Post) while Italy has liberalized the postal sector, i. e. besides the incumbent operator (Poste Italiane) there should also be a few smaller operators. THis means that in order to get a letter from Italy, to the pope, it must be handed over from Poste Italiane to Vatican Post. Assuming that the arrangement you described is still in place, this would raise the question how those letters would be billed on the wholesale level. Usually, when you send a letter abroad, the operator of origin has to pay wholesale postage to the destination operator; However, for certain services, there are billing arrangements that derogate from this rule:
• For postage due and paid response items, wholesale postage is billed in reverse so the destination operator has to pay.
• Cecogram items and war prisoner service are free on the wholesale level.
Now the question is, if the arrangement still exists, how is it billed on the wholesale level? Possible options would be:
• Poste italiane treats those items as International Postage Due. In this case, it would be free for the sender but the pope has to pay.
• Poste italiane eats the fees and hands the items over to Vatican Post as postage paid. For this to work, Poste Italiane would have to slam a “Port Payé / Italie” rubber stamp on each pope letter. (Without the rubber stamp, Vatican Post would treat the letter as postage due)
• Poste Italiane and Vatican Post have a special bilateral agreement to exchange pope letters on a cost-neutral basis similar to cecogram and POW items.
If it's the first option, there may be no legislation on the italian side at all governing free pope letters because international postage due is a regular service.
In the third case, we would have to look for a bilateral arrangement betwenn Poste Italiane and Vatican Post.
If it's the second option, the legislation we look for could be either an Italian law or a bilateral treaty.
This is a too-long comment, posted as an answer
The Pope and the Italian authorities finally sorted their relations in the 1920s. This lead to the signing of the Lateran Pacts in 1929.
According to Gordon Ireland, in "The State of the City of the Vatican," published in the American Journal of International Law, the Vatican and Italy signed a postal convention on June 29, 1929. Gordon says that in addition to setting up the Vatican's postal service, the convention provided that,
"correspondence from and to the Pope is to pass in Italy under the frank..."
Since the War, various provisions of the Lateran Treaty have been ratified, modified or rejected. I was not able to find anything about this particular clause.