For example, suppose I claim to have found a one-way portal to another universe. Or maybe it just annihilates anything put into it, instead. The claim that it's a portal is unfalsifiable because no one can send information back to indicate whether or not it worked, but if that portal is the only way to escape from something bad, then I care very much whether it works or not.

Somewhat off topic, but that all instantly made me think of this. I may very well want to know how such a portal would work as well as whether or not it works.

WARNING: Wikipedia has spoilers to the plot