If your phone has activation lock, you will have to get control of the old Apple ID to remove the account. The device should keep working, but you won't be able to sign in with the new account until the old is under your control.
You can read the details and what to do with a forgotten password at:
- iCloud: Find My iPhone Activation Lock in iOS 7
If you don't have activation lock, then open the settings app, and tap iCloud
Swipe up so you see the bottom of the screen and press "Delete Account" - answer what you want about any content and you'll know if you need the password when it's ready to remove your old account.
Also, it's not very easy or often even possible to bypass Activation Lock since that would defeat the purpose when someone steals the device.
- How to bypass iCloud login for iPhone 5 running iOS 7 when email and password was forgotten?
- How secure is Activation Lock on iOS 7 (and later)?
If you're not on iOS 7 or later, then please edit that into your question as you might have other options.