Is it irony if two men are speaking of war, and one of them complains that he is so tired that he is "nearly dead"?

Not really. Irony would be if one of the people in the conversation were, say, an ardent pacifist, losing an argument about “why war is bad” ... and then punching the person arguing for war because he makes a better argument.

Not irony, as @CyanoticWasp has said, but only coincidental (though not accidental, I’m sure).

David is using exaggeration or hyperbole to make a point; or perhaps he IS nearly dead.

Irony; The Gift of the Magi, by O’Henry. Husband sells his gold watch to buy his wife combs for her beautiful hair; meanwhile, she has cut and sold her hair to buy her husband a platinum watch fob.

An old man turned ninety-eight
He won the lottery and died the next day
It’s a black fly in your Chardonnay
It’s a death row pardon two minutes too late
And isn’t it ironic… don’t you think

Mr. Play It Safe was afraid to fly
He packed his suitcase and kissed his kids goodbye
He waited his whole damn life to take that flight
And as the plane crashed down he thought
“Well isn’t this nice…”
And isn’t it ironic… don’t you think

A traffic jam when you’re already late
A no-smoking sign on your cigarette break
It’s like ten thousand spoons when all you need is a knife
It’s meeting the man of my dreams
And then meeting his beautiful wife
And isn’t it ironic…don’t you think

Ironic – Alanis Morissette

I agree with @CyanoticWasp. @gailcalled cites one of the best-known classic examples of irony in literature.

Something that’s simply contradictory or contrary to your hopes isn’t irony, no matter what Ms. Morissette thinks.

Thank you for your answers!
I am glad I asked to confirm, I was debating whether to call it ironic in my essay. A lot of people in life call something “ironic” when it really isn’t, and I didn’t want to fall into that trap.

Not ironic. Coincidence.
Irony is a guy who nearly escapes death several times in a war only to die from slipping on the soap in the bathtub.

That Alanis song is terrible.