Some old footage of myself going around various channels and experimenting with what happened the first morning without Wii Connect 24. And btw, this was from 6/28/13 if you were wondering.

A few notes:
1:25 The news channel displays news from its last update which occurred at midnight.
3:30 Same applies for the everybody votes channel as news channel. Statistics and most recent votes are still displayed however you obviously can't submit a vote.
8:55 Keep in mind, this was the very last day before my Wii officially stopped reading my Brawl disk so the on and off errors that I've been getting are displayed.
10:55 As shown, I was able to send a replay to my friend gus. He did end up getting it later that day proving that at the time you could still send messages. When I got my Wii back the date had been changed to October 2008, and I was able to send a message via Wii Connect 24. This leaves me wondering if changing the date allows you to use Wii Connect 24. That is an experiment for a later date, though my hopes are low as I'm sure it would have been discovered by now if it does work.