Close a circuit with no current load (MOSFET)
There must be some current flowing or the switch/relay will have no effect. You should measure the current flowing when you close the switch. 

If you want to use a MOSFET then you need to check the voltage across the terminals when the switch is open, but I strongly expect that this will be a sufficiently low voltage that almost any MOSFET will meet the voltage range requirement. And, yes, MOSFETs can be used to switch very small currents. That's why we build computer chips with them.