Any way for VZ Guide and On Demand to work without VZ Internet?

[sathackr]

I'd like to use a different internet provided but maintain my FiOS TV service. Is there any way to do this while still maintaining functioning TV service? Right now my guide does not work and on demand doesn't work.

I've connected the Verizon router to the new internet provider, the internet light is green and I can access the internet through it.

[Edg1]

Yes, but you can’t use the Verizon router with the new provider. Even if you cancel FiOS internet, the Verizon router still has to stay in place to provide Guide/On Demand. 
You will have to buy another router for your new internet provider.