This has actually been researched extensively in socio-linguistics. In some parts of the U.S. (the Mid-Atlantic, mostly), there is no difference in spoken "Murray" vs. "merry". The data also shows a disparity in people's opinions of how well they pronounce the words. Most Philadelphians who don't make a distiction in the sound, for example, will judge their own pronunciations of the two words to be distinct even when the interviewer judges them to be identical.
Much of the confusion in this particular environment (where there is an 'r' in the coda (i.e., the part of a syllable that comes after a vowel)) is due to the peculiarities of 'r' in English. There are no other European languages with an identical sound, and the only other widely spoken language that has it as a coda is Mandarin (where it is comparatively rare). The tendency for a speaker of English is to reduce a short vowel that comes before an 'r' in the coda, and in some cases this leads to the complete absence of any segment other than 'r' (this makes it the syllable peak).
To answer Atlas' question, I pronounce it "burr-eed" (your provided phonetic transcription) in casual speech, but I wouldn't take speech advice from a Philadelphian, even a Philadelphian linguist
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What you were saying sounding a lot like the article, so I figured you either had to have read it, or been an expert in it 
