The Fallacy of unprovable evidence

I always forget what kind of fallacy this is. Its extremely common where any evidence shown is brushed aside is it just a moving the goalpost? What's the actual name of this fallacy it might be a group of fallacies.

1.Ancient Rome didn't exist all shown evidence and Primary sources were forged or made up. 

2. Evidence of Vaccines is forged by companies with a secret agenda

3. The earth is flat those pictures are just CGI.

This is close to the conspiracy theory . It is virtually always the result of people who use bad methodology to determine what is true. For example, with vaccines, people are trusting YouTube videos, Fox News entertainers, and social media posts from randos rather than world health organizations, medical institutions, and virtually all doctors and researchers. If the conspiracy theorists attempt to justify their position, they almost certainly resort to cherry picking by referencing the handful of studies or doctors that support their ideologically-based position (confirmation bias) rather than the thousands that don't. Psychologically, people crave being part of the elite few who "know the truth," despite the efforts of "Big Brother" trying to brainwash all us "sheeple."

The earth is flat. The universe is 6000 years old. Climate change is a hoax. Vaccines don't work. Trump won the 2020 election. Despite mountains of evidence to the contrary, these few Google University grads have discovered the "real" truth.

Wiser men posted the answers, and I have learnt the answer from them. However, I wish to present a different point of view.

I hope that this question is result of a curious mind, not a loaded question. I say this because of the given second example - Evidence of Vaccines is forged by companies with a secret agenda, and the original post is about people brushing aside every evidence or fact for a claim.

This sentence assumes that all vaccines (the way English language works 'vaccines' means all vaccines, without any exception), the ones that are strongly established to be effective, and the ones which are still being in experimental phase should have the same acceptability to the rational mind, and therefore whoever raises questions about the second category must be irrational.

In this time of pandemic where there is a debate going on for and against vaccines, this is a sensitive matter, and hence should be dealt with carefully.

People who can't think rationally about the given three topics, they are not capable or not willing to understand that they have committed a fallacy.  So I would argue based on only burden of proof.  So these scenarios can be seen as -
   Mr A claims something, and provides facts to support his claim. Mr B doesn't care about his facts, he dismisses the claim.
This is a matter of how a person examines presented facts.

This is an ideal scenario. But what if Mr A cherry picks his facts? In that case when Mr B dismisses Mr A's claim, one can't call Mr B as irrational, or conspiracy theorist.

The fact that governments worldwide call anyone who calls out their harmful agenda as conspiracy theorist, or mad, has caused fear among public that truth is being silenced, and the public can't do anything about it because they are helpless. (If you never have experienced incidents where you know something to be true, but government controlled media is saying otherwise, either you live in a wonderful, moral society, or you keep your eyes and ears closed.) This is a real issue. Fallacy of conspiracy theory is a delicate thing and should be handled accordingly.

When a senator posts such video about a case against Pfizer vaccine trial, it causes people to trust COVID vaccines less. It should be pointed out that no medical cross-checking was done in that clip. But when a senator posts such information, it is usually believed that every cross checking was done.

Laws are mostly, if not only, made to prevent bad people from doing harmful things. If that is not the reason, it is usually to protect the interest of few people in government/power. All other vaccines are not mandated, heavily encouraged only. But COVID vaccines are being forced on people, despite each of them being in experimental phase, rather than doing health campaigns, encouraging people. Not taking vaccine is not a harmful act, the people who want the vaccines to be fully tested before they become vaccinated are not bad people, so vaccine mandate does not look good. 

Then there are Pfizer's terms and conditions; also read the article with reference published by 'Public Citizen'. What you see here is Pfizer putting profit over humanitarian crisis. It should be noted that these are news articles, not articles or papers published in a peer reviewed publication. Two problems with peer reviewed publications are that - 1) not everything gets the attention of such a publication, 2) such a publication is controlled/checked by few people which can be easily manipulated compared to mass media. If I am not wrong, the risk of truth silencing is similar to risk of suffering loss in the stock market, the risk needs to be spread out, not put in a bottleneck situation.

Trust is a fragile glass.

Now, a second possibility, what if Mr A isn't cherry picking, but Mr B claims that Mr A is cherry picking? Now if there is a valid or invalid reason, the scenario is dismissed either way. If there is no reason at all then it is probably a combination of ad hominem and conspiracy theory- 'I don't like his facts because he is bad', and 'he is bad because he is powerful'.

An unprovable claim is a fallacy called unfalsifiability. 

When evidence is brushed aside, that is committing the cherry picking fallacy. They ignore inconvenient evidence that contradicts or falsifies their claims and only focus on cherry picking the evidence that supports their claims. Also known as conformation bias seeking. 

The 3 examples you give would be committing the conspiracy theory fallacy. 

Logical Form:

A is true.

B is why the truth cannot be proven.

Therefore, A is true.