Rye, as far as I know, warning message will only pop up when you edit case fields and tried to close the tab without saving the case. This message does not apply to chatter posts and emails.
However, you can save your chatter posts and emails as drafts.
Draft posts save automatically as you type a post. Once you’ve added text or a rich text item to your post, Saving... appears next to Share and then changes to Saved. When your draft is saved, you can access it from My Drafts. See full documentation here.
Same goes with the Emails, but it will not be saved to My Drafts. The text inputted in the Email Editor will instead remain in the editor even if you close the tab of the case. See full documentation here.