I've been flashed on these over the years many times when I am positive I did not run a red light, and have yet to see a ticket...

If you get cited they will send a photo of you/your vehicle to you to prove it was you (I've heard some people get a photo that is not them and not their vehicle and then they dispute it). Basically the citation is attached to the license plate and is the responsibility of whoever owns the vehicle. In the case of Avis if the ticket comes through as a red light run during your rental period they will contact you about it and probably send it to you with an admin fee for themselves added on.

Since the ticket does not attach to a specific driver I'm not sure your insurance will ever find out about it but I could be wrong there.

Be sure to check whatever address you have on file with Avis for possible communication (if you have a work profile or something- and a work address- check your Avis profile online).