Quiescent power of BJT vs MOS transistors
That isn't exactly a true statement "i.e. why doesn't it happen with MOSFET's ?".  It is True for logic type circuits, but for analog circuits both types of transistors consume power, and indeed in BiCMOS (the combination of the two) BJT's can be used to get performance at lower power, in some cases.

By restricting the answer to only logic based circuits we can come up with the following:

BJT's essentially operate on currents and the transistor (in a simple model) is a current amplifier.  CMOS's equivalent simple model is that of a voltage controlled current source, with the control gate being a capacitor.

In CMOS circuits, the transistor sources or sinks current onto the control nodes of other transistors "downstream" so the transistors's "current " aspect only sees capacitor loads (other transistors gates).  With a capacitor you only need to flow current to change the voltage on it, either by injecting current onto the node to bring it high , or draining charge off to bring it low.  Once the capacitor is at a given state, the current need not and indeed does stop flowing.

In BJT circuits, you must provide a current (albeit very tiny) to the next down stream transistor in order for the transistor to amplify that current to set it's new state.  It is simply the fact that you always have to supply some current in all states that BJT's consume more power.

To summarize (using simple models):

CMOS flows current on change, BJT flows less current continuously.