Excrement; shit. I have always thought of this word as having a strong Irish association, though the OED is silent on the subject. In the U.S., it is used exclusively metaphorically, that is, to mean that something is worthless, offensive or contemptible. One definition on Urban Dictionary recognizes shite‘s slightly euphemistic quality, calling it “the best way to say ‘shit’ without getting told off, as you can simply say that you were trying out being Irish for the day. ”
And [former New York Times Editor Howell] Raines all but comes out and writes that his predecessor, Joseph Lelyveld, produced shite. (Jack Shafer, Slate, March 24, 2004)/“We’re getting hammered by everything right now,” says Bert the Bug Man (a.k.a. Bert Bertrand, a licensed exterminator). “Smoky brown tree roaches are coming out of the sewer lines to hang in people’s foliage because people are trying to keep their plants alive.” So we’re not producing enough shite in this drought?? (Houston Post, blog post, June 21, 2011)
notoneoffbritishisms.com 
I’m an American but was introduced to this term during foreign exchange training with the Royal Navy (1987). My favorite usage heard then, after understanding the meaning of “shite”, was the term for a sea gull (which in the US I’ve heard called “rats with wings”)’ to wit: shite hawk.
Scottish dialect as well as Irish I believe, certainly I’ve heard glaswegians use it as the regular form of “shit”
Defintely agree with Urban Dictionary as per it’s use as a more polite expletive. My saintly mother, who “tried out” being Irish her entire life, used both “shite” and “arse” liberally, and did not discourage their use by her own children, even though she would never in a million years use regular foul language.
I’m not sure if it’s related, but in the Irish language, adding an ‘e’ changes from the positive degree to the comparative degree i.e. tall, taller.
Some examples in Irish are; fada (long), faide (longer); láidir (strong), láidre (stronger).
“Shite” is pretty much interchangeable with “shit” in Liverpool (UK), with the use of the diphthong giving an opportunity for extra emphasis. Often heard at the football: “you’re fucking shite you [insert name]!”
As has been said, in the UK it’s pretty much interchangeable with “shit”, although there are some phrases, such as “bag of shite” (descriptive of anything you don’t like – “Windows 8? Bag of shite”) where you *wouldn’t* alternatively say “bag of shit”, although you *can* say either “pile of shite” or “pile of shit”.
One of my favourite uses of “shite” is in the phrase “that’s like the difference between shit and shite” – used to state one’s belief that two options are as bad as each other. Most often used with similar-but-different things like politicians of opposing parties.
I never knew it was considered more polite than shit, in Ireland. I suppose it is like “feck”: it always amazed me that the UK TV sitcom Father Ted could get away with regular, gratuitous use of what was obviously just a different pronunciation of fuck, when actually using fuck would have been unacceptable.
As a Londoner joining the Britiah army in 1962 I only heard ‘shite’ from people from north of Birmingham. So: Liverpool, Manchester and points north, as well as Ireland. Since then it seems to have spread south
As I noted in your article on ‘arse’ pronunciation indeed the spelling ‘shite’ is more Irish; of all my mates only the one Irish spells it that way. OED however is not silent on it that I’m aware of:
‘NOUN & EXCLAMATION
British vulgar slang
another term for shit’
It’s also pronounced differently of course.