What is the term to describe a person who answers a question with irritability?

'not suffer fools gladly' is a common idiom that could be what you are looking for.

The phrase is not entirely directed only to responses to questions but could also be a way to describe a person's characteristic manners things like like an ill-founded requests or proposals that try to demand a response.

Irritability might not be precisely on.. although there is often some of that... Curtness, or flippant response or types of dismissive responses might be other ways to describe this sort of person's typical reactions to things they deem nonsensical or illogical on the surface.

suffer fools gladly at dictionary.com

A person who does not “suffer fools gladly” is one who does not tolerate stupidity in others.

or another:

not suffer fools gladly at Oxford Dictionaries

Be impatient or intolerant towards people one regards as foolish or unintelligent.

‘he was a perfectionist who didn't suffer fools gladly’

I heard the same broadcast in the car, it was actually "Says You", and since said word reminded me of my (comic) self, I thought I would remember it but promptly had forgotten what it was by the time I got home. I spent hours last night trying to find the word or 09/30/17 Says You podcast to no avail, and now it's driving me crazy! The word contained gnash, and sounded like one of those Old English insults...oh my goodness, I just looked up Old English insults and there it was! GNASHGAB!

An 18th century northern English word for someone who only ever seems to complain.

I, too, intend to use it frequently, mostly in reference to myself! So there you have it. Enjoy!

"As usual Mr Corncobb answered our query very acerbically!

acerbic [vocab.com][1}

harsh or corrosive in tone