It's not going to self-combust or anything, it'll just become a proper ornament on account of you not being able to use it. In short, consider this your final warning - update your Kindle by tomorrow, or it won't be able to access the internet to download anything anymore. To clarify, without this update from Amazon, you will not be able to sync your Kindle to the cloud, or access the Kindle store to buy stuff, or use other Kindle services on your device.
If you regularly use your Kindle, don't panic. According to cnet.com: "Kindles will update automatically via Wi-Fi, but if the device has been turned off or out of battery for a while, charge it up and make sure you Sync and Check for Items. When the updated has been applied you'll find a confirmation letter called "03-2016 Successful Update" on your device. You can check for it by viewing all Recent items in your Kindle Library."
Amazon have issued their final warning regarding the update...
As you can see, there are certain devices that require the update and those that don't. Those that don't include...
Kindle Paperwhite (6th and 7th generation)
Kindle 7th Generation
Kindle Voyage 7th Generation
Those devices that DO require updating are...
Kindle 1st Generation (2007)
Kindle 2nd Generation (2009)
Kindle DX 2nd Generation (2009)
Kindle Keyboard 3rd Generation (2010)
Kindle 4th Generation (2011)
Kindle 5th Generation (2012)
Kindle Touch 4th Generation (2011)
Kindle Paperwhite 5th Generation (2012)
If you're "too late" updating your device, you will be greeted with the following message...
"Your Kindle is unable to connect at this time. Please make sure you are within wireless range and try again. If the problem persists, please restart your Kindle from the Menu in Settings and try again."
Despite all the WARNINGS and articles circulating regarding the WARNINGS, there is another way you can update your Kindle if you miss the deadline: "You'll need to manually download and install the required update." THE HORROR!
More info on that via Amazon.