Is this an already identified fallacy?

RatWiki calls this the on-the-spot fallacy. The fact that a person doesn't have specific examples of phenomenon X doesn't mean phenomenon X does not exist, isn't happening, isn't important, etc. It just means the person doesn't have specific examples at that time.

Someone might not have these examples because they forgot, aren't an expert, or don't have the resources on-hand.

It's also usually an attempt by people to discredit another speaker, by seizing onto the fact they couldn't provide said examples. Person 1 could be losing the debate - wrong on every count - but take advantage of one lapse of knowledge by Person 2 to act as if they're a clueless bumbler, or are incapable of even having a viewpoint.

So we have a couple of invalid inferences here:

P) Person does not have specific examples of X

C) X isn't happening (non sequitur)

and

P) Person does not have specific examples of X

C) Person has no idea what they're saying (poisoning the well)

Also, if the person demands one recite technical, but tangential resources in order to have an opinion, that's also a red herring.

Now, it must be said that the inability to provide specific examples does not necessarily mean that phenomenon X is false...however, it does not also mean it is true. There are phenomena that should leave obvious traces of themselves behind if true...furthermore, there are phenomena that are highly unlikely based on the available/currently-known evidence. Lastly, there are people who are known to peddle lies. For these three reasons, sometimes it is okay to reject a claim because someone couldn't provide examples. Just make sure you're doing it within reason.

I think Rationalissimus answered this well. I would just like to add a situation where this is not fallacious, but reasonable. I don't watch Hannity, so I don't know if he actually claims a "win" due to the lack of examples. If someone were to make an argument that is based on opinion , and citing examples is the best way to proceed, if no examples are given the argument should be postponed until an example can be provided, or the argument should be ignored. Nobody wins here—this is just a refusal to argue knowing that the argument will not be productive with one or more concrete examples. For example:

Person 1: Your mom is a mean person.
Person 2: I don't think she is. What does she do that makes you think this?
Person 1: I can think of anything specific... she's just mean.
Person 2: Think of some examples, then we can address those examples. Until then, we'll just have to disagree.

The inability to provide even a single example is not proof (at least in the above example), but a strong indicator that the person is repeating a narrative, that is, they are repeating what they have been told and can't justify how they came to that conclusion. In other words, they have likely reached the conclusion by accepting what someone else has said.

In some cases, the claim is almost a truism, like "politicians lie". In casual conversation, not having examples is fine. But if you are making an argument, be sure to have an example ready—your strongest example. Yes, it does look bad if you can't provide a single example to back your claim. No, it doesn't mean your claim is false. In my opinion, it does mean that you have not adequately prepared to argue your point and you are probably wasting my time.

I don't think this is a fallacy because it is not an error in reasoning. 

In the example you gave, you mentioned Sean Hannity was prepared with examples to support his position , but the callers are not prepared with any examples.

If a person has laid out their reasoning behind having a certain position and  is confident in their understanding of the issue then is challenged on that, the logical place to begin the discussion is to ask the person challenging that position to offer facts or examples that support their opposition. If none can be provided then there is no reason to take that person's claims seriously. 

If we assume a debate is won by the person who makes the best case for their side, then it does mean the person who was not prepared with any supporting points lost the debate. That's why it's important to be prepared with facts and reasoning before taking up a position in a debate.

It can be argued that it's fallacious to publicly insist Hannity is wrong while being unable to provide any reasoning. I think this may qualify as proof by assertion—the caller is asserting a position without any information that would justify it but the assertion itself. And although Hannity can offer examples to support an alternative position, the caller nevertheless insists that their position is correct without ever providing proof.