I currently have a couple checking/savings accounts that are sitting at $0 because I found better accounts and switched banks for some reason or another, is it worth the time to try to close these accounts? Are there any drawbacks to keeping them open?
The main concern I'd have is that something will happen to the account while it's unattended. While you may not have any money in it to risk, you could have a fraudulent check written against it that causes you to incur NSF fees.
Your bank also might change its no-fee policy (I assume these are no-fee accounts, or there's an obvious drawback). If it does, it's possible you might not notice, and again then the fee might be assessed, overdrawing you and causing additional NSF fees.
The security concept of minimising attack surface could be stretched to apply here, especially if closing the account would mean the end of your relationship with that bank. Essentially more routes into your finances or personal information means more opportunities for fraud, more accounts to keep an eye on, more logins to remember/store, and even more paperwork/idle cards to check (for unexpected activity and T&C changes), store and eventually shred.
However I had a couple of online-only savings accounts with zero balance for a few years, at a bank where I have other accounts, and I didn;t worry in the slightest. (You can open the accounts online but have to phone to close them and sitting on hold is too much of a chore for me. Eventually they realised their mistake, brought in a minimum balance requirement, and after giving notice closed accounts with less that that in them)