If you're certain there is no mail to be kept in the old account it can be deleted.

Press Alt or F10 in case the menu bar is hidden. From the main menu bar: Tools - Account Settings Select the account to be deleted in the left pane. Select 'Remove Account' from the 'Account Action' drop-down menu at the bottom.

My apologies, I do not want to remove the account. My website hosts my email. They made some major revisions and the server settings were updated/changed. I had not been using Thunderbird for awhile. When I downloaded the version I have, I entered the new server information when I set up the email accounts. The email accounts did not change, only the host server information. Thunderbird pulls my emails without issue. Frequently I will get an onscreen notice that Thunderbird cannot connect with the old server settings. It has not impacted my email, only acts as a nuisance. Is there a way to remove the old server settings or am I destined to simply see the notice periodically?

Try this - in Tools - Account settings for the old account, uncheck the boxes for Check for messages on startup, and check for messages every X minutes... and anything else that might prompt to connect.

Frequently I will get an onscreen notice that Thunderbird cannot connect with the old server settings.

Can you post a screenshot of the error? https://support.mozilla.org/kb/how-do-i-create-screenshot-my-problem

Also, please post your Troubleshooting Information. At the top right of the Thunderbird window, click the menu button , then select Help > Troubleshooting Information. Press the "Copy text to clipboard button" and paste the information into your reply.