Demanding impossible evidence?

Traveling to Japan is hardly an impossibility or even that unreasonable.

Swimming in Fukushima's contaminated water might be prohibited, but certainly not impossible. 

People travel to Japan frequently, but it is not necessary to do so to validate or discredit your claim.

The fact is there are much more precise scientific means for measuring the safety of Fukushima's waters other than actually swimming in it. 

Discrediting your claim by challenging you to swim there is just a weak argument and red Herring, It doesn't really prove or discredit anything even if you were able to pull it off.

Now, saying: "I'll believe something when pigs fly" would definitely be an appeal to the impossible.

Uhm, I think this is very good and it depends a lot on the reasons why they don't want to accept the sources you offer them. Do they know that the sources in Fukushima are not safe because of the testimonies of the people who live there or because of some source further away from the origin? If it is the former, then they have good reasons not to believe in the sources you offer them, if it is the latter, then they simply do not have good reasons to reject the sources you offer them. Certainly the second is a bias against trusting the media, perhaps it is a conspiracy or something like that.

f you presented only sources, then you committed the fallacy of the Appeal to Authority.

If you presented facts and used the named sources as the origin of your facts, then “Them” can examine and challenge the facts and the underlying method of concluding they are facts.  

Let’s take a look at the text of what you have: “You can post all the sources you want, I will believe X when you do Y.” This statement is not unreasonable. Therefore, the discussion then revolves around the reasonableness of Y. For example, he might have said,  “You can post all the sources you want, I will believe X when you do Y (present evidence to establish the conclusion).” 

You do not present the issue of certainty here, but I find it useful to keep in mind our goal is to establish “epistemological certainty,” not “absolute certainty.” 

 This bias is called RAISING THE BAR FALLACY

This is a specific variation of Shifting the Goal Posts.

If you keep shifting the goal posts so far that it is impossible to see the goal posts anymore then you get the RAISING THE BAR FALLACY.

Where the evidence and the proof required is deliberately made more and more difficult to accomplish.

Similar questions have been posted in the past. Short answer, there is no ideal matching fallacy (I can think of), it is a combination of cognitive biases and perhaps some denialism that results in people rejecting adequate/reasonable evidence. It is an overall inability to properly evaluate evidence.

However, I have added this to my list because I do think it might be common enough to justify a fallacy (perhaps Appeal to Unreasonable Standard). I would broaden the fallacy to address unreasonable standards, not just impossible, because I would not want to deal with the philosophical issues surrounding impossibility.