what is an Arduino Funnel
an use a different battery other than a Li-Poly battery for the Arduino Funnel
Yes you should be able to power that from any battery. However charging a Li-Poly battery is a bit more tricky than a lead acid. See:-
how does that affect the operation of the Fio?
Shouldn't affect it at all.
if I just have a carbon zinc battery, 3V or 6V, touching the li-poly battery terminals, the Fio won't go on.
Why are you doing that, never connect batteries together as they try and charge / discharge each other. Also you mess with the source impedance which is probably stopping the voltage regulator from working.
Should I be putting the input voltage somewhere else in order for the fio to turn on?
no.