Are there any FCC restrictions on broadcasting a test nav signal on 108.0 in the air? Or perhaps on other nav frequencies that can be received within the navigation band?
The answer is yes with the caveat that there are limits. The hard part in dealing with aviation radio systems is that there are multiple agencies with some overlapping regulatory authority.
As bogl points out, the ITU is the international forum for radio spectrum. The NTIA is the US agency that is the US representative to the ITU. The NTIA has general oversight of radio spectrum. Actual management of radio spectrum is delegated to various government organizations.
The FCC is the primary US manager of spectrum, but not the only one. The FAA and the military both have significant pieces of spectrum that they manage. For the FAA, these include the aeronautical radio navigation bands and the aeronautical radio communication bands. The VOR/ILS are within the VHF radio-navigation band of 108.00 to 117.95 MHz.
The FAA controls transmitting of navigation signals. They do allow for low power transmissions for the purpose of testing avionics. But you just can't build your own transmitter unless you're willing to go through a ridiculously lengthy approval process that includes both the FAA and the FCC. The FCC is involved because any transmitter has to comply with Part 15 for non-interference.
So, if you are using an approved piece test equipment you can use it for short periods of time to test avionics. The use of nav/comm ramp testers such as the IFR-4000 is quite common.
The use of radio transmitters is strictly controlled almost everywhere in the world. FCC would be the authority in charge if you are asking for the USA.
The International Telecommunications Union (ITU) is in charge of allocating frequency bands for different applications in the radio spectrum. You are only allowed to transmit if you have a frequency assignment, if your equipment is certified, and you are holding the required radio license.
There are use cases where non-certified equipment may be used (e.g. amateur aka HAM radio), or no license is required (e.g. wireless consumer devices like cell phones, remote controls, etc).
108.0 MHz is at the upper edge of the frequency band allocated to FM radio broadcasting, and the lower edge of civil aviation frequencies. Therefore 108.0 MHz is not available for experimental navigation signals. Your usage would be illegal. The same will be the case for any other frequency that can be used with certified navigation receivers.