If the account uses IMAP, then any action which ends up with a message being removed from a synchronized folder in the client should, at some point in time, result in that same message being removed from the IMAP server. Ordinarily, moving a message from an account folder to Local Folders in Thunderbird will achieve this.

Caveats:

• Whilst a message may appear to have been deleted, the implementation details (at the server) will determine whether it is deleted immediately, deleted later (maybe as part of garbage collection), or never, and perhaps just hidden. This last one is a common scenario with gmail's IMAP; deleted messages remain unseen in All Mail, not appearing in other folders simply due to the tags system.
• Users who have chosen to use the experimental maildir storage in Thunderbird find that deleted messages mysteriously re-appear. (This will be irrelevant to the vast majority of users who employ the default mbox mail store)
• Some smartphones allow you to delete locally yet leave messages on the server (e.g. Blackberry.) This may be relevant if you wanted to remove a sensitive message by use of a smartphone or tablet; that message may remain visible to regular email clients such as Thunderbird

POP is simpler; by default it moves a message from server to client, deleting the server copy as it does so. Exceptionally, an email client can usually be set up to leave a copy on the server, so it then falls to the user to manage the accumulation of messages on the server. Most POP servers seem to offer quite limited storage, so letting messages accumulate leads to the server becoming full quite quickly.

…if it is set to delete deleted files from the server…

Thunderbird might be set to ask the server to delete the message, but that's all it can do. How and when (or if!) the message is deleted is entirely dependent on how the server manages such requests. As I mentioned above, some servers are quite reluctant to really delete things; they just hide them.

Thanks

Great answer