I have received a large number of questions regarding various instant coffee devices on pesach and shabbat. Here is a quick summary of the issues for those who wish:
The Keurig device has two sharp tubes which puncture the coffee packet. The upper one serves to inject boiling hot water into the cup, and the lower one serves as the point of egress for the brewed coffee. Both tubes are inside the boiling coffee at the time of cooking. Both have metal and plastic components and can not be kashered for Pesach.
However: If you have been careful to use only unflavored coffee throughout the year from the time of purchase of the Keurig, then your Keurig has always been (and continues to be) kosher for passover. In this case, the device does not even need to be cleaned, as it is already fit for Pesach use. You probably want to clean it anyways in a passover sink, but that's just to make sure it isn't gross. ;)
The countertop water heater ("kumkum") is even easier. Since it has only had water in it, it is also already kosher for passover. wipe down the outside and it is ready to go.
Boiling water is a forbidden activity on Shabbat (it is a "toldah" of "dye making" to be specific). Therefore any benefit derived from newly boiled water on Shabbat is also completely forbidden on Shabbat. There are 3 common household items that are called into question: the dishwasher, the "instant hot," and the water heater.
Dishwashers. Dishwashers that require a handle to close and lock them in order to function can only be loaded, not run, on Shabbat. Modern dishwashers with timers can be set prior to Shabbat, and loaded on Shabbat as long as the timer buttons are not manipulated on Shabbat. If it goes on later, great. If not, too bad. You may take out the dishes in the morning and use them for shabbat, but should not take out dishes that you don't need yet. Those get put away after havdalah.
Instant Hot. This device is installed at the sink, and boils the water as it passes through the spicket to your mug. There is no way to use this on Shabbat. Some have a hot water reservoir which automatically refills and reboils if you drain water out to use. This is p'sik reisha, a direct and inevitable forbidden action, so is prohibited.
Hot water heaters. This is the large tank or tanks in your home that provide water for sinks and showers. They routinely heat and turn off in cycles to keep a constant temperature. The water as a whole does not boil. Water at the metal surface of the heating implement within the tank may boil from time to time. As you drain water in your sink or bathrooms, new water comes into the tank and mixes with the much larger amount of existing water. The addition of new water does NOT force the tank to boil, as in the instant hot. Therefore it is not p'sik reisha, is indirect and unintended, and is therefore permitted. You may take hot showers on Shabbat, and you may even choose to use the hot water in your sink to mix tepid instant coffee (yuck).