Not near Rome —not even close!—, but since I was planning a (since cancelled) Italian trip not that long ago, I did a bit of research on the subject.
The Palazzo delle espozioni often has photography exhibits, When will you be there? There's a Don McCullin retrospective starting in October:
Don McCullin in Rome
www.palazzoesposizioni.it
The Officine Fotografiche Roma looks like an interesting place devoted to photography, and they do hold exhibits:
Officine Fotografiche Roma è un’associazione non profit riconosciuta nata per divulgare e sostenere la cultura dell’immagine. L’Associazione è da sempre luogo d’incontro e di scambio tra professionisti e appassionati, un centro di formazione per fotografi a tutti i livelli, un laboratorio...
roma.officinefotografiche.org
It's closed in August. If you go later, you could email them. They would know if there are interesting exhibits elsewhere.
Hope you get more info from actual Romans...