My take is that if you don't use them and love them, don't keep them. I do have DVDs and even some VHSs (of movies that have not been put on DVD), but they're not the run-of-the-mill movies that you can easily rent either (I'm an oldies fan). Even of those, I keep only what I enjoy watching more than once.
You might be able to get a little money for your DVDs, depending on what they are. I don't know if you'll get anything for VHS tapes - even at thrift stores they seem to be a glut on the market.
When my children were at home, I had a specific place for movies. This was back in big-bulky-VHS-land, but the movies could not overrun that space. When they did, I would ask everyone to participate in sorting time. One child would look through all the movies and put aside his "can't live without" ones; the next would go through the remaining stack and set aside the ones she had to keep - and on down the line. My husband and I would do it, too. We ended up keeping only the movies someone had a declared liking for, and I disposed of the rest.
However, the main issue is how to work it out with your BF so that your action is a team one. Your children are watching you to learn how adults make decisions together.