By default, FogBugz deletes emails from your email server after successfully downloading them. This applies to all mailbox types (POP3, IMAP, EWS).
For IMAP you have the option to disable email deletion in FogBugz Mailbox settings.
For POP3 and EWS, depending on your email service provider, you might be able to configure to keep the emails on the server, by overriding the removal setting with your service provider.
POP3 and EWS
For POP3 and EWS, overriding this setting can be done only if your email service provider allows this. Please check with your email service provider how this can be done.
For example for GMail this can be overridden as described here: Troubleshooting - Emails are deleted from GMail.
For IMAP you have the option to disable this in FogBugz Mailbox settings, by removing the checkbox as shown in the screenshot below: