I could be wrong but I don't believe the Pantheon is managed by the church, even though it technically functions as one. I believe it falls under the jurisdiction of Polo Museale del Lazio, which is a branch of a larger cultural office. A rough, google-translated version of the mission statement from its website (http://www.polomusealelazio.beniculturali.it) states:
"The Museal Polo of the Lazio region was established under the D.P.C.M. August 29, 2014, n. 171, in force since 11 December 2014. The Poli Museali Regionali, offices of non-general management level, are peripheral joints of the General Directorate of Museums. Provide in the territory the provision of the public service for the enjoyment and exploitation of the institutions and places of culture to be delivered to the state or state entrusted to management, including those relating to the institutes referred to in Article 30, paragraph 2, letter (a) and (3), by defining common strategies and objectives for enhancement in relation to the spatial scope of competence, and promoting the integration of the cultural paths of fruition and, in conjunction with the regional secretary, the resulting tourist- cultural."
http://www.polomusealelazio.beniculturali.it/index.php?it/232/pantheon
That said, I don't have an issue with a few euro to see very, very old churches with fantastic art/architecture, pre-Christian ruins under the the foundations, etc. even if the funds go to a religious organization tasked with their upkeep. I figure it's fair for adding my own wear-and-tear to their fabrics?
Shoot, given the recent news pieces on the alarming deterioration of Notre Dame in Paris, I wonder if an entry fee might be order there as well? (I belong to no specific religious affiliation).