How do I Remove old password from App Store? I made a new Apple account and the old Apple account/ password has to be used to update apps.

Those apps will ALWAYS require the Apple ID they were downloaded under. Apps are FOREVER tied to the Apple ID. If you don't want to have to use that ID, you'll need to delete the apps you downloaded under that ID and repurchase them under the new ID. That's why it's generally a bad idea to use a different Apple ID. If you have an issue, you can change the ID Password, and keep the ID forever.