The word originates from Latin fŭror, -ōris (accusative form fŭrōrem) which is the source of present-day French fureur (pronounced /fyʁœʁ/; a homophone in French to "führer") and Italian furore (pronounced /fuˈrore/).
Many French words with -eu- in the last syllable have variants with -ou- in other dialects or older forms of the language. I don't actually know if that's the case for this word, but it seems to have entered English with the spelling "ou".
The earliest OED citations are for a derivative of the French form:
1477 Caxton tr. R. Le Fèvre Hist. Jason (1913) 29 Considerest thou not the strengthe & force of my body and the furour of my swerde?
1489 Caxton tr. C. de Pisan Bk. Fayttes of Armes iii. xxi. 219 A madde man duryng his fourour may not be reputed nor taken for enemye.
1509 A. Barclay Brant's Shyp of Folys (Pynson) f. lxxvv Where..wrath doth rayne with his furours.
1525 Ld. Berners tr. J. Froissart Cronycles II. xlvii. 162 Some oppressed..with the furoure of the see.
a1542 T. Wyatt Coll. Poems (1969) ci. 1 What rage is this? What furour of what kynd?
In Middle English, "ou" in words from French is thought to have been pronounced /uː/. In monosyllabic words, such as hour and flour, Middle English /uːr/ generally corresponds to modern English /aʊə(r)/, but in the disyllables like savo(u)r and labo(u)r the stress ended up on the first syllable, and as a result the vowel in the second syllable is not pronounced as /aʊə(r)/ (compare adjectives ending in -ous, which have a reduced vowel in the last syllable).
The spelling "-or" in words like this is at least partly from Latin influence; it may also have been facilated by the stress pattern mentioned in the preceding paragraph (Noah Webster notoriously promoted the use of the spelling "-or" in place of word-final unstressed "-our", even in some words that were not from Latin like neighbo(u)r).
Maybe in part due to the influence of the "-or" spelling, the word furor is commonly pronounced in modern English with an unreduced "o" sound in the second syllable. The OED entry for furor (first published 1898) gives "Brit. /ˈfjʊərɔː/, /ˈfjɔːrɔː/, U.S. /ˈfjʊˌrɔ(ə)r/, /ˈfjuˌrɔ(ə)r/". But pronunciations with a reduced vowel are noted in other dictionaries: AHD "(fyo͝o´rôr´, -ər)" and MW "\ ˈfyu̇r-ˌȯr , -ər \".
The Italian form has also been taken into English, however. The OED's first citation is from 1790, with more from the 19th century:
1790 E. Wynne Diary 15 Feb. (1935) I. ii. 34 Went to the opera... They made a great furore for Mrs Banti.
1831 J. C. Young Diary 16 June in Mem. C. M. Young (1871) I. vi. 208 I heard Paganini. The furore there has been about this man has bordered on fatuity.
1851 T. Carlyle Let. 10 Sept. in Coll. Lett. T. & J. W. Carlyle (1998) XXVI. 163 This blockhead..is..making quite a furore at Glasgow.
1864 W. Lewins Her Majesty's Mails 263 It was little thought that..they would excite such a furore among stamp collectors.
The OED entry for furore (first published 1898) gives the pronunciation as "/f(j)ᵿˈrɔːreɪ/ /f(j)ᵿˈrɔːri/ /ˈfjʊrɔː/".
Obviously, there has been some influence between these theoretically distinguishable words.
The spelling furore and the associated unique pronunciations appear to now be considered chiefly British. Collins has a "regional note" saying "in AM, use furor"; the American Heritage Dictionary defines furore as "Chiefly British Variant of furor".
The pronunciation variant /ˌfjʊ(ə)ˈrɔː/ that you noted in Oxford Dictionaries is interesting. I would guess that it comes from interpreting the final "e" in the spelling of "furore" as a marker of stress.
Can "furor(e)" = "führer"? Yes, but some people don't like this pronunciation
Whether or not furor sounds the same as führer/fuhrer/fuehrer depends on the pronunciations that you use. For the latter word, MW gives " \ ˈfyu̇r-ər , ˈfir- \" and Collins gives "E ˈfjurər ; fyo̅orˈər" which overlaps with one of the possible pronunciations of furor.
However, some prescriptive peevers like Charles Harrington Elster (author of The Big Book of Beastly Mispronunciations: The Complete Opinionated Guide for the Careful Speaker) apparently think the pronunciation of furor with an reduced vowel is substandard (or in Elster's words, not a "cultivated pronunciation"):
furor FYUUR- or, not FYUUR-ur.
Most English nouns ending in -or are pronounced like those ending in -er. Thus, actor, factor, governor, juror, sector, vendor, visitor, and so on, all have the same -ur sound at the end as maker, officer, seller, thinker, and the like.
Furor, however, is an exception, for we pronounce the -or to rhyme with more, not her. Exceptions are sometimes impossible to explain, and the best explanation I can offer is in this case is that FYUUR-or has been the cultivated pronunciation since dictionaries began recording prounciation in the 18th century. Webster 3 (1961) was the first dictionary to recognize the -ur variant. It does not appear again in any of my dictionaries until M-W 9 (1985) and current dictionaries all list FYUUR-or first — Encarta (2001) prefers it. The NBC Handbook (1984), playing the maverick, is the only authority that prefers -ur.
Elster goes on to suggest that the pronunciation of furor with a reduced vowel may actually have been influenced by the pronunciation of Führer, but I'm somewhat skeptical of this explanation. In any case, Elster concludes the entry by saying
Let's try to keep the German separate from the English: FYUUR-ur for führer and FYUUR-or for furor.
So I guess the Charles Harrington Elsters of the world might consider it "wrong" to pronounce furore the same way as fuhrer.