It depends.

If you are using IMAP then they are on the server and you will not have access.

If using POP then they are on your local drive but associated with your email account. If you tell Thunderbird to remove the account it will do just that including the messages.

Make folders under the Local Folders account in Thunderbird and move the messages there. Once you have them moved and you are sure you can read them and have not just copied the headers then you can safely do away with the old account.

You might want to Google IMAP vs POP so you understand the differences. People that do not understand the difference between the 2 protocols are usually surprised by their actions.

Thanks for the useful information. If I go the local folders route, will I also get any attachments associated with the e-mails loaded to the local folders? Also: Any other solution you can think of, aside from using an e-mail client program, to keep all my old e-mails (including attachments) from my internet provider's server since I'm going to be closing that ISP account and probably opening another? thanks very much.

thebrainsta said

Thanks for the useful information. If I go the local folders route, will I also get any attachments associated with the e-mails loaded to the local folders?

Also: Any other solution you can think of, aside from using an e-mail client program, to keep all my old e-mails (including attachments) from my internet provider's server since I'm going to be closing that ISP account and probably opening another? thanks very much.

If you do really want to save old emails outside your email client, the ImportExportTools add-on is quite useful for exporting from Thunderbird in a variety of formats.

https://addons.mozilla.org/en-US/thunderbird/addon/importexporttools/