If you weren't paying a monthly rental for the old router (like you do for any set top boxes you have for your TVs) then you probably own the old router outright. They used to include them in some manner within the install kit/fees. I think you are the proud owner. Congratulations.
If you are renting the new router ($10/month or something like that), they might want it back if you were to discontinue service. If you bough it outright ($149 or $199 depending on the offer you got), it's yours. Congrats again in that case.
Keep it up and you may be able to start your own ISP.
If you didn't explicitedly purchase it , its best to hold on to the router in case they want it back. They have been known to insist on there return when you terminate your service even for OLD routers.
In the mean time you can use it for various things such as extenting your network to remote areas of your home.
Hi! I had the exact same quesiton, and so called Verizon just now. I was told that yes, it must be returned within 30 days. I told her the old router does not fit well in the box the new one came in, and she said to fit in the best I could. There was a return label in the box that had the new router. I also have the option of returning it to a store, but NOT every store accepts returns of equipment, and you can't tell by looking on the website. I had to wait for her to find one based on my zipcode.
I hope this helps!