No, it’s pork as you know it (heh) with a tiny extra bit after the usual vowel. Like you hiccuped in the middle of the word. Pork does not have that sound in RP. It may have had in some other accents, but not in RP like the Wiki claims.
So kind of like “po-work”, but much faster and smoother?
Pretty much.
The ɔ is the sound you say - if you speak SAE (Standard American English, also called General American English, or GA) - in hot. The ə is the vowel sound at the end of words like water (followed by an r in SAE). Basically, if you’re hit very briefly in the stomach, and make a sound, that’s the ə. There’s no “w” but there can be if you really emphasise the word.
If any accents ever did/do say fork like fɔək, as the Wiki claims (it does sound like some accents I’ve heard in really old movies and there might well be some accents that do it still), odds are they would also say pork, etc, the same way, so it still has a rhyme.
I must say I enjoy this site. Before I discovered SDMB, I never would have found myself enunciating pork/fork over and over while drinking my morning cup of coffee.
I feel like the Swedish Chef. 
Don’t rhyme to me - I speak English with a mild Scottish accent. Unfortunately I’m no use at doing IPA symbols to demonstrate the difference, but they are different.
American with a slight Southern accent. Despite all of the crazy Southern accents and diphthongs I’ve heard in my life, pork and fork have always rhymed. Even with the explanations given above, I can’t figure out how to pronounce it so that it doesn’t rhyme.
English with a weird mix of South African/Estuary accent here, and they rhyme.
How does the first set not rhyme with the second set?
I heard a woman from Atlanta in a documentary who pronounced ‘war’ as ‘WO-wah’. Two distinct syllables with a ‘w’ in the middle. Now that was a crazy Southern accent! 
Well, to me, the vowel sound in “pork” is the same as the one in “oar” or “board”. The sound in “fork” is…uh, different. Maybe like in “clock”. I really should learn some IPA. Or record myself saying the words, but I don’t have anywhere to post a recording.
Huh, this is fascinating. I have elderly relatives who I remember saying something that sounded like “po’erk” for “pork.” This would be in upstate NY. I always chalked it up to their general accent, which I would describe as hick.
Ohhh, Bjork, Bjork, were you brought by the stork,
Or were you created from butter and cork?
I love you so much that I act like a dork,
Oh Bjork, oh Bjork, oh Bjork.
– The Bjork Song, The Brunching Shuttlecocks
Came to mind because of the bit later in the song: “What’s the next rhyme?” “um…da da da…pork.” “Pork? How does that fit in?” (singing) “Let’s go to my bedroom, that’s where we can…” “I GET IT! I get it…”
Has anyone come up with an authoritative cite for this supposed alternate (or “primary”?) pronunciation? It’s all sounding like a case of Whippy-wanking to me.
Both my RH3ED and my Shorter OED give a single pronunciation: rhymes with dork, as in “Wikipedian.”
Sorry, I don’t understand. Are you disputing the fact that people with different accents pronounce different words in different ways?
They rhyme with a Yiddish accent.
No. I’m asking if there’s any cite at all from an authoritative source that “pork” is pronounced any second way by any significant population. So far, it’s Whack-o-pedia against the world, including the OED.
I’m not sure why people are confused.
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Pork and Dork clearly rhyme.
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Fork and Spork clearly rhyme.
The things from list 1 clearly do not rhyme with the things from list 2.
Cork, however, rhymes with both lists 1 and 2.
I don’t know if Scotland counts as a “significant population” to you, but I’ve already said that I pronounce those differently and I’m not unusual among people with my accent.
Can’t give you a cite - on this one you’ll just have to take my word for it.
For whom? Again, I have two authoritative dictionaries that show identically rhyming pronunciations for all four words, and no alternatives. What region, dialect or population has differing pronunciations?
I’ll certainly accept your word for it, but it would seem that somewhere there’s a cite. Simply saying there’s an arbitrary difference among anything but a very minor group (such as a city neighborhood) only goes so far.
There are of course many regionalisms - “warsh” cloths, “idears” etc. but this is the first I have ever heard that there are different rhyme classifications for common words ending in “-ork.” I’m obviously not alone. Some substantiation besides What-If-O-Pedia must be out there unless it’s a truly microcosmic phenomenon.