I can't begin to count the number of theists who have said to me "you can't prove god doesn't exist". As far as they are concerned this lack of falsifiability is carte blanch to believe anything they want about "supernatural", unfalsifiable "things".
In truth, however, this lack of falsifiability is one of the most basic reasons the burden of proof is on the believer. If a lack of falsifiability were sufficient grounds for accepting a proposition as true or respectable, then every nonsensical fantasy would have to be accepted as true or respectable. UFO abductions, fairies, leprechauns, the Loch Ness monster--all true by that logic!
So, if you hear this ridiculous claim, you can say:
"That's exactly why the burden of proof is on you--because it is literally impossible to prove it's false."
Then you can explain why they already know this--because they apply the same rule to those who believe in things they don't believe in.
You can also follow up with one of my favorite responses:
"The complete lack of evidence for a god meets whatever burden of proof I might have."