Short answer: get a bigger hard disk.

You could disable synchronization and free up the space on your local disk without loosing the messages. However, this would eliminate the possibility to create backups of your messages.
Since you mentioned this is important business mail, you shouldn't take chances.

Hi "christ1" - thanks for your reply. I can't get a bigger hard disk as I am using a tablet PC :-( Yes it's business mail but keeping the sent items is more in case I ever needed to go back in time than needing to keep a backup of everything for auditing purposes etc., so I'm prepared to take the risk that the service provider who is hosting the mail server is taking their own responsibilities seriously and backing up their own systems! So - the next question is, assuming I disable synchronisation, how do I then delete sent messages stored locally without losing the originals on the remote server? Having just deleted a message from TB that wasn't important, I can see it has also been deleted from the remote server. Thanks in advance.

First you deselect the folder from synchronization.

Tools menu (Alt+T) > Account settings > Synchronization and storage > advanced

Then you delete the mail in the folder because It is not synchronized anymore. (try your experimental delete first)

Unsubscribe from the folder. Right click the account name in the folder tree and select subscribe. and deselect the sent folder

Hi Matt - I couldn't get this to work. I have synchronisation switched off, then tried deleting a mail. It disappeared from the server as well. If I unsubscribe from a folder, that folder disappears from the account in TB, which is not the behaviour I'm looking for...

I presume you are using IMAP.

when you say 'local storage of Sent emails ' do you mean:

A.) these emails are stored in a 'Sent' folder in 'Local Folders'

or

B): do you mean you have syncronised the server 'Sent' folder for offline use and so have a copy on the computer, but when you delete an email from that folder, it resynchs with server and thus removes off server?

If you mean (B) the folders are one in the same. Just one copy is stored on your computer but it is designed to update with the server when you make changes, so if you delete an email then it will also delete off server.

Whilst you might need access to old sent emails, do you need to see all of the Sent emails all of the time?

To save disk space you could try the option of only downloading xx emails or xx days of emails...see here: https://support.mozilla.org/en-US/kb/imap-synchronization#w_configuring-synchronization-and-disk-space-usage

You might find the option of not selecting to synch all messages and folders, but only selecting the folders you need to see more appropriate. see section at the above link. At the top of the page, Keep messages for this account on this computer should be checked if you want to synchronize messages between your local system and your mail server.

Another option would be to reduce the size of the 'Sent' folder by Archiving old sent emails in another folder. So putting old sent emails into a different folder which you do not synchronize. Then the Sent folder will be smaller and if you need to see older sent emails then you select to synchronise with that folder, view email and then unsych the folder.

IMAP synchronization for TB means an offline copy of every message is being downloaded to your local disk. This is why you do have a large Sent folder.
It does not mean that you can delete messages locally, and that this change isn't reflected on the server. So you must not delete messages you want to keep.

As you stated in your OP you still want to access messages in the Sent folder, so do not unsubscribe from the Sent folder.

I suppose you already did disable synchronization in the settings for your IMAP account.

Before freeing up the space on your disk I urge you to create a backup of your Thunderbird profile folder.

Once you have a backup, locate your profile:
Help - Troubleshooting Information - Profile Directory - Open Directory

This will open the profile folder in your file manager. Change to ImapMail and find the folder for your account.

In the account folder find a file Sent (without a file name). There should also be a file Sent.msf.

Close Thunderbird.

Now delete the file Sent. This is the local copy of your messages on the server.

Start Thunderbird.

As stated before, you now don't have a possibility to backup messages sent and received from now on. If for whatever reason something gets lost, it's gone.