If you have a fireplace with a remote ignition controller, this project may be of interest. On the geek-tips website, they've described how to modify remote-ignition controllers with Ethernet-enabled Arduino boards so they can be easily turned on or off via a connected device. Like most devices with simple button control the modification is simple, and with the teleduino system you could even do this from almost anywhere. However it would be wise to consider the unwanted possibilities of such as system - you don't want any "unrequested" ignitions!
If you're looking for an Ethernet-enabled Arduino-compatible board, consider one of Ethernet-enabled boards, such as the Freetronics EtherTen - the Arduino Uno-compatible with onboard Ethernet, microSD socket and optional PoE: