Please tell how do we correctly pronounce the word February. Though common pronunciation should be Feb-RU-ary, in some dictionaries it is Feb-U-ary where the first "r" is dropped. Which one is correct?
This question could be much improved if you gave specific mention of the dictionaries you've checked.
My Mac's online dictionary clearly shows both pronunciations are acceptable:
Click on the speaker icon at Dictionary.com, and you will hear both pronunciations, separated by an "or":
With so many reputable sources listing both pronunciations as valid, the word could be pronounced either way. If you manage to find a dictionary that only lists one pronunciation (such as Collins), that would indicate some abridgement.
For the record, here's how the OED lists it (both with a silent r, and an enunciated r):
You already know that there is no "correct" pronunciation. The Longman Pronunciation Dictionary (2008) gives seven variants used in British English and two variants used in American English.
As for the word "February", it's a famous example of the divide between BrE and AmE. The most common pronunciation in BrE is with "ru" and in AmE it's with "ju". According to a poll conducted by Professor Wells, the variant with "ju" is becoming more common with the younger generation in Britain; cf. "Feb-yoo-" is now becoming the accepted standard (Oxford Dictionaries Online).