Basically, the form is serious limericks - but limericks in which the last line deliberately doesn't rhyme. Here are two examples:
To novels and plays not inclined,
Nor aught that can sully her mind;
Temptations may shower,
Unmoved as a tower
She quenches the fiery arrows.
Religion makes beauty enchanting;
And even where beauty is wanting
The temper and mind
Will shine through the veil with sweet lustre.
Really, it's poetic equivalent of having your teeth drilled. I think what makes it particularly annoying, apart from the moralising - if you're going to moralise, you need to make deeper observations than that - is that the final line may not rhyme, but it does scan. You can break with the form completely and have a quite pleasing effect, for instance (borrowed from this site):
The limerick, peculiar to English,
Is a verse that's hard to extinguish.
Once Congress in session
Decreed its suppression
But people got around it by writing the last line without any rhyme or meter.
That isn't annoying, because once the line runs on past the last syllable the ear relaxes, knowing that the form is being properly broken with. It's as if we were running a race and came in second: we don't get the satisfaction of the tape breaking across our chests, but we run a few paces and cool off. But when the unrhymed last line scans, it's as if we were running a race and the judges sneakily replaced the tape with a brick wall. Thud.
I remember an interesting lecturer at college talking about how the limerick is so inherently comic in its sound that it's difficult to contrive one that isn't funny (or at least, doesn't feel like it's trying to be.) His best example was on Sir Walter Raleigh:
Sir Walter was handy with cloaks,
And tobacco, and packets of smokes.
Such a mighty romancer
Of insomniac cancer -
I thank him, and hope that he chokes.
But I wouldn't say that was successfully serious either. A limerick might well lend itself to anger, but it comes out feeling like an epigram, and epigrams are another form with comic overtones.
I can't think of other non-comic limericks either. There are some that aren't funny if you take them seriously, so to speak, such as Edward Gorey's:
To his clubfooted child said Lord Stipple,
As he poured his postprandial tipple,
'Your mother's behaviour
Gave pain to Our Saviour
And that's why he made you a cripple.'
- which is very upsetting if you think about it, but animated by Gorey's dark humour; it tends to produce an appalled laugh. Edward Lear could do something similar, made less funny by his outmoded tendency to repeat his first line:
There was an Old Man on some rocks,
Who shut his Wife up in a box:
When she said, 'Let me out,'
He exclaimed, 'Without doubt
You will pass all your life in that box.'
Which is pretty creepy, really, and has that uncomfortable diminuendo that Lear's repeated last lines tend to have, but it still feels comic in its form. A lot of Lear's comic poetry is minor-key and curiously sad, and this is no exception, but you wouldn't call it a serious poem.
So the Reverend Patrick Bronte seems to carry the laurel for unfunny limericks, and he does it by - with the best of intentions, I'm sure - using the last line to thump you hard enough that you aren't amused. Unless anyone can think of another contender, I think it's Bronte in the lead.
Here's my take on the subject:
Rev. Bronte, a worthy old cleric,
Whose children wrote books atmospheric,
Tried verse for a time,
But his endings lacked rhyme -
An effect that is oddly frustrating.
Anyone else got one?