Can I remove old phone jack boxes?

Here is an article that might be helpful - https://www.hunker.com/13415235/how-to-remove-a-phone-jack - good luck

We also get our telephone service from our cable provider. On our setup there is a telephone wire that goes from the modem box to a telephone jack and back-feeds all of the jacks in the house. You need to find out if your jacks are live, try plugging in a phone to see if they work. If they are live perhaps you could just upgrade to better looking jacks. If you are determined to get rid of them it probably will not hurt to cut the wires flush so that none of the copper wires can touch each other. I would give it a couple of days here and hopefully an ol' phone guy will land on you question and send you off in the right direction.

Thanks. These jacks are really old. Approx 2.5 x 2.5 square mounted outside the wall or on the baseboard.

No, we have them too.

As long as they are disconnected from any power source... I simply ripped them out and patched the holes. They are low voltage so there is no issue with them even living in the wall with the boxs removed.

Sorry, I totally get it. I found another option. https://www.youtube.com/watch?v=XkCXe59YcEg

May be easier and you could maybe decorate the plate that goes over it with something like this ... http://www.hometalk.com/search/posts?filter=outlet%20plates

Unscrew the mounting screws on the box. Unscrew the wires coming out of the wall from the terminals. They are low voltage ( like two nine volt batteries) won't hurt you. Tape off each wire with electrical tape so they don't touch each other. Feed the wires into the hole and patch the hole.

Thank you, William! This is the kind of answer I was hoping for. One would think Verizon could answer that. Are you sure doing so won't interfere with those jacks I am using?

Thanks everyone! Trouble is boxes are drilled into baseboard thats 70 yrs old. Cleaned the up as much as I could. New owner will have to deal with it.