They shouldn't discredit any source at all.
One should not be able to ask a question excluding evidence, because that's a cherry picking fallacy. Similarly, we might ask the OP to present a verbatim version of a claim to avoid straw man fallacies.
For example, one should not be allowed to ask "I don't trust Project Blue Book (a long term study of UFO sightings which found no evidence of aliens), are aliens real?" in favor or the unbiased question "Has there been any sighting or contact with aliens?"
See: FAQ: Should questions be asked with an impartial tone? and Should I ask a question if I am already highly suspicious of the source?
However there is an implicit "lower limit" to the quality of evidence that is provided in the answers set by the source of the claim, if the asker provides one.
For example, if the asker posts a claim of a current event from a news media, it is invalid to answer by posting evidence from a blog or another news media, otherwise we simply end up with a "he said, she said" debate which helps no one. Conversely citing many independent media sources, or giving a historical perspective is probably fine. For example, many questions about Fukushima received terrible or no answers at the time, but some of them got higher quality answers over time.
Another example is sustaining the veracity of a claim by presenting repetitions of it. It doesn't work.
This leads to two cases of unanswerable questions:
Single source questions, such as fresh news. Usually these are single source and it is not possible to find any expert analysis of the evidence or historical context. It might seem very important and notable on the moment to answer these, but it's often impossible to verify any evidence until months or years have passed.
See: Handling news questions about current events. Prior to this meta question, we did allow current event questions such as the Fukushima ones due to this answer which led to the various bad current news questions you might find on the site.
Questions about the scientific status quo. While of course it is a valid question in itself, if one doubts a systematic study on the subject or something uncontroversial, we are unlikely to be able to find something better.
See: Are rebuttals of specific scientific papers on-topic on Skeptics Stack Exchange?