As far as I know the history of the plates of Regia Marina is the same of those of Regio Esercito. But the numbering system was different and it was used with no changes by the Italian Navy until 1986: the first digit showed the naval base where the vehicle came from, the second was always 0 and then the last three were the number of the vehicle (for more details about it, see the Italian Navy).
Guglielmo Evangelista wrote me that he found out that the vehicles that were requisitioned by Regia Marina during the Second War due to the increased needs had a plate with RM followed by a serial number starting from 1, but he hasn't got any clear picture yet.
The last two pictures show two plates for motorcycle of Regia Marina: they had a front plate mounted on the mudguard exactly like the Regio Esercito. You can also find the full picture I took the detail of the second plate from in the old photo gallery (picture 16).
Plates for motorcycles
A special corps of the Royal Navy was the Xª M.A.S. and it also had some special plates. Anyway I prefered to insert the documentation about them in the web page of the Italian Social Republic.
I must thank Roberto Pola very much for the first picture. The last two images are taken from the Ufficio Storico della Marina Militare and I want to thank Guglielmo Evangelista for sending them to me!