Idiom: People caring about minor stuff while something terrible is happening

The classic:

Rearranging deck chairs on the Titanic.

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.

@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."

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.

Recently heard the phrase:

He's making the beds while the house is on fire!

Which seems to be fitting for your question.

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.

"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.

There's this quote by Edmund Burke which neatly sums it up—

He pities the plumage, but forgets the dying bird.