berk

I had a query about the origins of berk "an idiot" too, with a suggestion that it might be related to birkie, a Scottish word for a man, particularly one who's a bit too full of himself or to the verb to burke, which is used literally with the sense "to murder" and figuratively to mean "to hush up; to avoid".
The OED has citations for birkie from 1724 to 1816, but the Dictionary of the Scots Language extends that to about 1930, with birkie also being used to refer to anyone with a quick temper, including women and children. These extended dates are just about late enough for birkie to be a candidate for the origins of berk and the semantic change from "conceited person" to "idiot" is reasonable, but you'll see that there's a stronger candidate.
To burke is from the name of William Burke, an Edinburgh grave-robber who was executed in 1829. The verb was used until at least the 1950s, but the semantic change and grammatical shift from "to murder" to "idiot" are less convincing. Without any evidence to support a connection, we can discount this as a possible origin for berk.
The OED has citations for berk, also spelt birk and burk, from 1929 to 1963, and a quick search on Google Blogs confirms my sense that it's still in use in Britain, though it's now a very mild insult. Please let me know if it's used further afield.
The rhyming slang origins for berk didn't seem terribly convincing at first glance (OED etymology: Berkeley/Berkshire Hunt = cunt), mainly because I hadn't heard of either, but it turns out that both Berkshire and Berkeley have hunts and that they're both jolly famous if you happen to mix in those circles. Berks has been used as an abbreviation for Berkshire since at least 1787 (which is the earliest example I can find in The Times), but Berkshire is pronounced Barkshire, in Britain at least, and its abbreviation is usually pronounced Barks. This means that if we want to make the case that Berkshire Hunt is the origin of berk, we need to assume that this rhyming slang is based on a less prestigious pronunciation (and this wouldn't be an unreasonable position). Members of the Berkshire Hunt call themselves 'Old Berks', but I haven't been able to find early enough evidence to suggest that this is the origin of berk "idiot" rather than a jokey reference to it. I also haven't been able to find any evidence that Berkshire Hunt was ever used in full to mean either "cunt" or "idiot", and I suspect that Berkshire Hunt is only given as a possible etymology in the OED because Julian Franklyn, an authority on rhyming slang, saw it as the more correct form. He mentions to burke when he's discussing berk, but only to say that they're unconnected and that it's necessary to distinguish between the spelling of the two.
The case for Berkeley Hunt as the origins of berk is made all the more compelling by the fact that Berkeley Hunt has also been used with the senses "cunt" and "fool" (OED dates: 1937-1977), sometimes in full, but also in the forms Berkeley and old Berkeley. The earliest citation for berk also includes the fuller form Berkeley Hunt. The OED entries for berk and Berkeley Hunt were written later than Franklyn's dictionary, so he didn't have access to this information.
In conclusion, then, the OED appears to be correct in tracing berk to Berkeley Hunt. This might  have been re-enforced by association with Berkshire Hunt later on, but it looks as if Berkeley Hunt is where the term originally came from.