After I deleted a Google account I had used for work, my Galaxy smartphone kept showing an 'Account action required' notification.
Even if I dismissed it, the notification came back after a while. The cause was that the account had not been removed from the phone itself.
So here is a simple way to remove the account from your phone.

Account action required notification

Open the phone's Settings, go to 'Accounts and backup', and select 'Manage accounts'.

Settings > Accounts and backup > Manage accounts

In the list of accounts (email addresses) on the phone, select the one you want to delete and remove it.

Remove account