I am not aware of other service providers that use the type of box/set up Verizon does. The cable companies set up the services the same way they do for tv service and just provide a tabletop modem to connect to the computer or wireless router.
I saw some of your other responses to posters and one mentioned reviews of Verizon you found online. I have heard some things about them but they have usually been because of pricing changing from statement to statement and customer service issues. I have a close friend that has Fios and he loves it. He didn't have a problem with the install and hasn't had an issue with service. He only has the internet option and not the TV or telephone package. The box was put in the corner of a small pantry room.
I would call them and create a "story" that you just moved into a home and have this ugly Verizon box mounted inside and you want to remove it and want to know if you will be penalized in any way. Either way, have the tenant be responsible for the removal. Their name is going to be on the Verizon contract not you so I think the equipment would be theirs to deal with financially just like a cablebox would be.