Impedance of a function generator when off
It depends on the details of how it is implemented. If the power is applied then output impedance is probably going to be 50\$\Omega\$ if the amplitude is turned down to zero, but if the output is turned "off" (with power applied) then there may be a series analog switch or relay that actually makes the output high impedance (not to be confused with the "High-Z" setting which really only affects the output amplitude). 

In the case of the Rigol DG4062, there is a series mechanical relay contact that disconnects the output when it is turned off or when power is removed from the instrument. Otherwise it is 50 ohms. I believe this is also true of the Agilent 33220A, but I can't verify it at the moment.