Update Facebook Status message using an Atmega16 microcontroller
This is half software, half hardware question.

From the hardware side, you'll need a way to get the ATMega connected to some Ethernet so it can connect to the Internet. There is no native support for Ethernet on an ATmega16, so you'll need to use a supporting IC that will do the Ethernet for you. These supporting chips usually communicate to another processor (your ATMega, in this case), over a simple serial link, such as SPI or I2C. As Garrett Fogerlie mentioned, you can take a look at some of the Arduino shields. You don't have to use an Arduino - just look at how the circuit works. 

The Arduino Ethernet shield uses a supporting IC that communicates over the SPI bus. So first, you should figure out how to get SPI working on you ATMega. After that, this question starts to tread into software territory quite quickly. As others have mentioned, the software side of this is likely less than trivial. I don't know anything about communicating with Facebook, so I'll stop here.