Imagine a situation in which the whole place is on fire, a bomb is about to explode, everyone is running for their lives and someone is checking his looks on the mirror... pretty inappropriate for the situation, don't you think? I'm looking for an idiom or colloquial/slang/informal expression to describe such an indifferent stance.
Copying from my comment to @Mitch's answer I think that rearranging the deck chairs is applicable in a scenario when someone tries to correct a doomed situation, cosmetically. In the scenario mentioned, I feel that “fiddling while Rome burns” might be a slightly more apt phrase.
Rearranging deck chairs on the Titanic.
@Mitch & @Jogabonito's answers are perhaps more apt, but someone who sweats the petty things while neglecting the larger problem may also be considered "penny wise and pound foolish."
Recently heard the phrase:
He's making the beds while the house is on fire!
Which seems to be fitting for your question.
In this instance I'd definitely go with Mitch's suggestion (Rearranging deck chairs on the Titanic) but as an alternative, someone who
can't see the wood for the trees is someone who focuses on small details but fails to appreciate the wider context.
In software engineering the term "bikeshedding" is used to describe this situation.
For more information read the Wikipedia entry for Parkinson's Law of Triviality.
I've heard straining at a gnat, but swallowing a camel.
I always liked the quote from the musical "The Music Man":
There's a burglar in the bedroom while you're fiddlin' in the parlor!
Seems to fit fairly well.
"Ignoring the 800-pound gorilla in the middle of the living room", or "failing to address" the same.
If the person's nonchalance in the face of danger is due to informed confidence that the threat will be taken care of by others, I'd say they are "cool as a cucumber" or "steely-nerved".
If, on the other hand, it is due to irrational denial or dismissal of the threat, I'd say they are "bonkers", "loony", or some other synonym for "out of their gourd".
If, on the third hand, their lack of response is due to distraction or inattention that is keeping them from perceiving the threat, I'd say they are "out to lunch", or "bliv"--a contraction of "oblivious" that has been used as a derisive appellation in my family for decades.