I would say yes they do, as 100.3(B) in the NEC states that listed or labeled equipment shall be installed and used in accordance with any instructions included in the listing or labeling. I'm positive that the manufacturer of the exit signs has testing instructions and testing requirements for the equipment.
Look at it this way, an NFPA 110, Level 1 system (NEC 700) is allowed up to 10 seconds to transfer the required emergency loads to the emergency source and restore power. The unit equipment within the exit sign will function as the emergency source of power for the exit sign from the loss of the normal source of power to the transfer to the emergency source of power. The exit signs will always be energized
An additional thought to take into consideration, if there are any emergency luminaires that provide emergency lighting for the exterior of the exit doors, be sure that these emergency luminaires are not supplied with emergency power by the exit sign unit equipment.
I'm late to the party but I have some input. NFPA 70 (2017 version) Article 700.16 says that Emergency illumination shall include means of egress lighting, illuminated exit signs, and all other luminaires specified as necessary to provide required illumination.
NFPA 101 (2018 version) 7.10.5 goes into detail regarding illumination of signs. My interpretation is if there is egress lighting that will light the sign until your back up generator picks up the load you are not required to have an internally illuminated sign at that location thus not required to test it. However, if you have an EXIT sign mounted in a location that would not be visible upon loss of power you need to provide means to make it visible at all times. There are battery back-up versions, glow in the dark versions, and internal power source versions (some kind of low grade nuclear source).
Point is, just because you have a battery back up version mounted that doesn't automatically mean you are required to test it. In the locations that don't have emergency lighting you could mount photo luminescent or nuke version.
It is probably unintended but it is actually a trick question. The term "emergency" generator is used very loosely in our industry. I notice that there is no code Article referenced anywhere in the question. In order to properly answer, we first need to know if it is a true Article 700 generator, or an Article 702 generator. Many times when unit equipment is used in conjunction with a generator, it is because the generator system doesn't comply fully with Article 700 and/or NFPA-110. So the net effect is that the generator becomes an Article 702 generator. More about this here:
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