When you get your new passport, you’ll usually be given your expired passport back, as well. The National Passport Processing Center will typically mail your old one back with your new one, clearly marked expired, and if you get your passport at the post office, they’ll probably let you keep your old one.
The question is… what do you do with it now?
If you want to keep it for sentimental value, go ahead. However, you’ll want to keep it somewhere safe. If a thief breaks into your home, what you see as something you keep to remember those fun adventures you went on, they will see as an opportunity for identity theft or they may even sell the passport to a forger.
If you want to get rid of it so that you don’t have to take the risk, you’ll want to shred it or otherwise destroy it. The last thing you need is somebody digging through the trash for credit card numbers and so on, and finding your passport there.
Whether or not to keep it is up to you, just make sure it stays out of the hands of criminals, whether that means shredding it or keeping it tucked away in a safe or a locked drawer.