iPhone: Disable syncing for mail account, but keep existing e-mails
MS Exchange is a system not too different from the way IMAP works: most of the mail is kept on the server and generally only recent mail and/or headers are kept on the local PC.

You can keep your mail but it is a bit of a work-around. Essentially you need to create a bunch of new mailboxes that mirror your folders on your Exchange account. You would go to

Mailbox &gt; New Mailbox... &gt; Location &gt; On my Mac

Give the folder a name and repeat for any other folders you want to save, making sure that when you create the new folder it is "On my Mac"

Once that is done, open the Folders of the Exchange mailboxes one at a time, select all the emails (Command A) and drag them to the correspondingly named folder you just created. repeat for all folders.

Remember that Exchange may not have downloaded the whole email and you may just be dragging the email headers from one mailbox to another. Without a connection to the Exchange server that is the best you can do.

Now you could delete that account without fear of losing your email.

It is possible to get an archive of your email account, assuming the Exchange server admin is willing to or allowed to do so. On an Exchange server you can export a mailbox or whole email account to a PST file. Once you have the PST file you can use Outlook for the Mac (or a free conversion utility) to open the PST file and view your old emails.