Nope, just natives "talkin' quick" and too lazy to add the ly. No rule applies here.

Do it quickly!
The adverb "quickly" describes the action "do".
You could say: Do it! Quick! Those are two separate thoughts.
You will also hear: Do it quick! That's grammatically incorrect.
The basic pattern is:
She spoke seriously.
He waited nervously.
I drive carefully.

Some verbs are followed by adjectives, not adverbs.
Be serious!
She got nervous.
They are becoming more careful.
These verbs often relate to states rather than actions.

With other verbs, it depends on the meaning.
He looked carefully at the test results.
They look good together.

[edit] Hi Selkie! I must learn to type more quickly. :-)

"Do it quick" is perfectly grammatical (like "drive slow", discussed here earlier) -"quick" can serve as an adverb as well as a noun. (Check your dictionary!)

Glad to:
Search here on "quick"
http://www.oup.com/oald-bin/web_getald7index1a.pl

Scroll to the bottom and read:
WHICH WORD?
Quickly is the usual adverb from quick....
Quick is sometimes used as an adverb in very informal language, especially as an exclamation: Come on! Quick! They'll see us.

So I'll stand by what I wrote.

[edit]
Of course "quick" is also a noun. What's that got to do with the question?

@SD#: you wrote "You will also hear: Do it quick! That's grammatically incorrect." Do you still stand by that assertion?

I don't have a strong opinion on 'quick' vs. 'quickly'; when both are possible, 'quick' is more colloquial/informal, 'quickly' more formal/standard.

But to me, 'serious' is in a different category. 'Stay serious' is correct, because 'stay' is a linking verb and the complement is an adjective, like 'Be serious.'

With an active verb such as 'take,' however, you must always use 'seriously.' Lines from pop songs such as *'Take it serious' are not just informal, they're wrong. Substituting 'serious' for 'seriously' is a real (and glaring) mistake, significantly worse than a casual colloquialism such as 'Do it quick!'

The use of "quick" as an adverb goes back a long way, though it may now be mostly informal and it may currently be the fashion to regard it as uneducated or sloppy.

People who do use it may make a distinction in meaning or usage.

Most people who would say "Quick, do it now", "Do it quick". "Get rich quick." would not say "He tied his shoelaces quick" or "He quick threw the ball . . .".

So I don't think it's laziness. Perhaps they are trying to retain the English of Shakespeare, Milton, Tennyson, Dickens, etc., and the others are too lazy to make subtle distinctions.

@Martin: Yes, I obviously know people say it, but I see it as grammatically incorrect. The OUP site I quoted calls it "very informal." I don't know what the OUP site's definition of "very informal" is, but given the intended audience, learners of English, I wouldn't be particularly surprised if it was a flag to warn learners that they are likely to get points deducted.

There is no single authority on the English language. As the language changes over time, what once was wrong may become accepted. In English this is a gradual process. We do not all jump at the same time.

In addition, professional writers often allow themselves greater freedom (one might even say "liberties") with a language than, say, a learner of that language can afford. Perhaps you are viewing this from the standpoint of a professional writer.

@MikeE: Perhaps they are trying to retain the English of Shakespeare, Milton, Tennyson, Dickens, etc.  
Yes, absolutely! Of course, that's what they are doing. *g*

"Quick, do it now!" is "(Be) quick, do it now" and adding an -ly would be quite wrong.

There are differences in usage of course: He tied his shoes real quick is heard in many parts of the US. (Zounds, who knew so many people harkened back to the poetic days of Shakespeare and the like?)

Please explain the difference in meaning you see between "Do it quick" and "Do it quickly" and I will try not to be so lazy as to overlook it.

@SD#, yes, I agree, "get rich quick" is less formal than "get rich quickly". And I agree that it is important to flag that to those that are learning English. But I don't see why we would to label an informal usage as "ungrammatical".

"He wants to get rich quick" is informal. *"He wants quick rich to get" would be ungrammatical. The distinction is worth preserving.

@Martin: You and I apparently draw the dividing line in different places. (I don't say "drive slow" either, although I know that some people do.) Since there is no ultimate arbiter, we will have to agree to disagree.