Must authors of non-fiction tell the truth?
I'd say yes, writing non-fiction is intended to convey knowledge of something. Knowledge is justified true belief. This does not necessarily mean the knowledge conveyed is ultimately true, just that the author has a justification (which can be examined) for believing it is true.
For example, when the ancient Greek's wrote about astronomy they wrote about their justified true belief; it just turned out to be a wrong.
This reminds me of a line from Indiana Jones: "Archaeology is the search for fact ... not truth. If it's truth you're looking for, Dr. Tyree's philosophy class is right down the hall."
Must readers who detect things that cannot be not true give authors the benefit of the doubt by concluding that what seemed like non-fiction is actually fiction?
No, I think that we need to respect authorial intent. If I wrote something I'd rather someone disagree with my points than extol my work as fiction.
We also need to be cautious when we claim something cannot be true. Is it that the claim runs counter to an underlying assumption or worldview we hold, or is there actually something objectively or logically wrong with the claim?
Where does religious or spiritual writing fit into this consideration?
There is no difference when it comes to religious works. A work of non-fiction in this area still contains justified true belief. As with any work, that justification can be examined and explored.