Which one is correct?
You heard it correct
You heard it correctly
Does the same apply to read it correct[ly]?
Correct is an adjective and correctly is an adverb. In Standard English, but not necessarily in other varieties of the language, verbs are modified by adverbs, so You heard it correctly is grammatical in Standard English and You heard it correct is not.
Correct (an adverb is needed to modify the verb heard or read):
What may be throwing you off is the common expression "heard you right":
In that phrase, "right" is an adverb, even though it sounds like an adjective.
The standard way of saying this is
There is a trend in many dialects of American English to allow the use of adjectives for adverbs in certain circumstances. This is one of those circumstances. If you think you're hearing "heard it correct" a lot from Americans, you are probably right.
If you click through, Google shows you roughly 500 hits for "heard it correct" and "read it correct", just as many as for "heard it correctly" and "read it correctly", and I presume that there are lots more hits for each of these phrases that they're not showing you. So this "mistake" is very common, although I am sure that "heard it correctly" is used much more frequently.
I wouldn't advise trying this at home unless you know how to do it right, because you may end up sounding absolutely terrible, even to American ears: