Is there a minimum mosfet drain current?
There is no minimum drain current.

However, the MOSFET has a leakage current which will flow even if it is off.  If your load is on with only 25nA, then you might have trouble turning it off.

Also, when you turn the MOSFET off, its parasitic capacitances still exist. So, if Vgs=0 from the MOSFET driver, you still have Cgd+Cds caps which have Vds=0V across them. Current going through the load will need to charges these capacitances, until Vgs reaches the same value of the power supply and there is now 0V across the load...

So yes, you might need to put a resistor across your load, or use a push/pull drive (ie, two transistors, one to power the load, and the other to short it to turn it off and discharge capacitances). If you use low voltage, the output of a simple logic gate will do.