"Don't mistake me"

Many times during conversation I hear the phrase Don't mistake me. Is it grammatically correct when used to mean Don't take negative connotation of my word. Shouldn't they say Don't take me by mistake?

Mistake can be used as a verb, exactly as you cite it in your example. The noun use is a lot more common.

Don't mistake me is perfectly grammatical, but it doesn't mean anything like "please don't interpret my words negatively". When mistake is used as a verb, it means "identify incorrectly" or "misinterpret". Thus, you can get usages such as:

Don't mistake me for my sister.

or

Don't mistake me: I'm not disagreeing with what he said, but he could have expressed himself more politely.

I think the phrase you may be looking for is "take amiss" (like FumbleFingers said).

Please don't take my words amiss, they were not meant to offend.