I pronounce err to rhyme with fur. I'm British.
I don't pronounce the h's after w's.
hth

I definitely pronounce err to rhyme with fur, and my feeling is most other Australians do to.
Australians don't usually distinguish between "w" and "wh". To my ears it sounds like a typically Scottish thing to do.
Hmm, looks like I'll have to stand corrected on the "fur" thing. Strange though because I don't remember ever hearing anyone at all say it, and I lived in the UK for 2 years. I am sure I am not currently exposed to anyone who talks that way.
Elinor Glynn was an early 20th century English author of rather steamy novels. She coined the term "It" as a euphemism for "sex appeal." A contemporary little verse was based on a scene in Three Weeks, published in 1907--
Would you like to sin
With Elinor Glyn
On a tiger skin?
Or would you prefer
To err
With her
On some other fur
Having taken that little side trip, I'll add that I (native Californian) say "to air is human."
I think i would say err to rhyme with fur.
though, to be honest i can't think of an occasion when I have used it other than the expression "to err is human, to forgive divine"....
always associated the "wh" pronounciation as scottish with the one exception being English Teachers. English Teachers as in teaching literature etc.

As an Canadian speaker from Ontario, I say "air" for "err", on the few occasions I would use it. How do people who say it like fur pronounce 'error'?
As for the wh sound with a pronounced h, I think I don't usually say it, but when my husband makes fun of me being outraged, he says "Whhhaat?", so I can only assumed that sometimes I do say it. Disclaimer, I have never been an English teacher.

I'm English, and I pronounce 'err' to rhyme with 'fur'. It had never occurred to me that people might pronounce it differently!
#15, bjd: I pronounce the 'e' in 'error' to rhyme with the 'e' in 'get'.
#14, sneaker fish: How about 'to err on the side of caution'? That's the only time I ever use 'err'.
I also pronounce 'wh' as 'hw', although I'm not and never have been an English teacher. However, I've got a bit of a posh accent. Everyone else I know (apart from my mother but including my kids) uses the flat 'w'.

I'm Irish and always pronounce whine differently to wine. It's in no way a conscious attempt to be correct, it simply feels natural. I think the consensus in previous discussions was that this is standard pronunciation in Ireland and England but rare in the rest of the world.
I say "err" in the same way as both "air" and "heir". If I say the verse quoted in #13, err rhymes with prefer but not her or fur.

Thanks to LaGrande for suggesting "to err on the side of caution". I couldn't decide how I pronounced this, since I hardly ever do. But trying that over in my mouth, I think I (New York, ancient) rhyme err with fur, her, and prefer.) Errant (same root, "wandering") and error get the vowel of merry which to me is different from that of air (=Mary).
I don't think I pronounce wh as hw unless I'm being very careful. My English-teacher father did.
Any first-year law students present may be interested to know that Elinor Glyn was the sister of Lucy, Lady Duff-Gordon, the defendant, and loser, in Woods v. Lucy, Lady Duff-Gordon, 118 N.E. 214 (N.Y. 1917) a Cardozo opinion likely to come up in Contracts (no consideration was formally specified, but the writing was "instinct with an obligation . . . imperfectly expressed" and so is a valid contract). That Lucy, by the way, is not to be confused with the Lucie Lady Duff-Gordon who wrote a wonderful volume of Letters from (upper) Egypt in the 1860s. and who, by the way, was the daughter of the philosopher of law John Austin. I think Lucie's husband was Lucy's husband's uncle.