Is the correct pronunciation Feb-ru-ary, like the spelling, or Feb-u-ary?

Answer 1

Unfortunately it is pronounced as FEB-RU-ARY. However, I expect that may change because hardly anyone says this.
It wasn't that long ago that the plural of matrix was "matrices" (also index and indices.) My brother made fun of me when I said "matrixes"when I was 12.
It's unfortunate that the pronunciation and usage of words in common parlance governs both their meaning and spelling. This is how words come to mean different things over time. the more a language is adulterated, the more it changes. The less it is used, the more pure or original it remains. As an example of dumbing down the meaning, see "awesome."
O tempores! O mores!

Answer 2

I tend to say Fe-bru - ary

I came across this will looking up something else. It's legit, unlike my other statement. I thought it was interesting.

[Middle English Februarie, from Latin Februrius (mnsis), (month) of purification, from februa, expiatory offerings, possibly of Sabine origin.]
Usage Note: Although the variant pronunciation (fby-r) is often censured because it doesn't reflect the spelling of the word, it is quite common in educated speech and is generally considered acceptable. The loss of the first r in this pronunciation can be accounted for by the phonological process known as dissimilation, by which similar sounds in a word tend to become less similar. In the case of February, the loss of the first r is also owing to the influence of January, which has only one r.

If you go to http://www.thefreedictionary.com/February they have a button to click where you can hear the pronounciation.

Answer 3

The latter