If only this question had come up in a few months' time - humour in the face of death is the focus of the chapter of my thesis I am working on, but it's in really early stages, so I probably can't give as full an answer as you would necessarily like right now. My work is focused on commemoration in epitaphs (and to some degree, funerary art) so that's where this answer will be focused. Others may be able to offer a broader take.
In short, yes, humour is a staple for dealing with death from a very early stage. To offer some Classical context for humorous approaches to death, the Milan Papyrus (dated between late third century, and early second century BCE) contains a number of epitaphs, some of which are collected in a section labelled ‘tropoi’, or ‘characters’, and are probably not for real individuals, but are comic exercises in the form. One epitaph demands:
Why have you stopped, won’t let me sleep,
And, standing near my gravestone, keep
On asking from what land I came,
And who’s my father, what’s my name?
The deceased then gives his name, father’s name, and home country (all details to be expected from an ancient Greek epitaph), before demanding that the passer-by at this imaginary gravesite walk on, declaring ‘We foreigners don’t like much talking’, playing on a stereotype of Cretan folk being a bit on the taciturn side. (Gordon L. Fain, Ancient Greek Epigrams: Major Poets in Verse Translation (University of California Press, 2010), p. 95 https://books.google.co.uk/books?id=1bz5wAPk1HsC). Clearly, humour can be used to characterise the dead and lightly parody our expectations of commemoration. Humans have been comfortable with making light of death for a good long while.
To turn to the Black Death more specifically, it's worth remembering that at this point, you had to be of decent means to get a grave monument in the first place, and many plague-dead were buried hastily in mass graves. I don't know of any particularly comical epitaphs for the plague-dead, though comic epitaphs are certainly not at all unheard of at this time (Guthke's Epitaph Culture in the West is a good source for this). That doesn't mean there aren't any, just that I haven't seen them - and if anyone knows of some, TELL ME!
Plague humour is certainly accessible in a broader social context though. The most famous of these is probably the Danse Macabre, or Dance of Death which was a pervasive theme in art and literature following the Black Death. In these images and poems, death is depicted coming for each level of society, from pauper to King, summoning them into one last, frenetic dance. The visual imagery here is often quite deliberately comic - for example, the Danse Macabre des Femmes (printed 1491) includes images of skeletal musicians, and beautiful young women engaging in courtly dance with visible rotting skeletal figures as their handsome beaus. (see here for the book on Gutenberg complete with illustrations). The figure of death is often laughing, teasing, and shown as 'catching out' senior figures in a way which can only be regarded as comical.
Staged drama also held elements of this bleak humour attached to genuine loss. Richard Beadle describes how the York Cycle of medieval mystery plays includes a representation of the plagues of Egypt, in which the loss of the firstborn child is described as the 'grete pestelence', the customary way to refer to the Black Death. Beadle describes the receipt of the news of the 'grete pestelence' as follows:
Upon hearing the grim words, the tyrannous and verbose Rex Pharaoh is immediately deflated, capitulates and orders the release of the Israelites - in performance, a moment of chill stasis after the pell-mell black comedy with which the reports of the preceding plagues of Egypt would probably have been presented. When the play was new, this my also have been a moment of remembrance for survivors of the Balck Death and those born in the succeeding generation, for the York Hosiers' play of Moses and Pharoah was probably composed when the memory of this, the most destructive of all the plague's visitations, was still a living memory. (The Cambridge Companion to Medieval English Theatre, ed. by Richard Beadle, 'The York Cycle' (pp. 85-108)).
While medieval and early modern folks certainly had a real taste for bleak comedy, comedy also had other, more structured purposes. In the Culture of Pain, David B. Morris describes how the Bible verse, 'a merry heart doeth good like a medicine' (Proverbs 17:22) was often treated rather literally, and comedy could be seen as '[strengthening] Christian readers for resuming their combat against the world, the flesh, and the devil'. (p.94). One of the most striking examples of this in literature is Boccaccio's Decameron, in which a group of people escape the Plague by retreating to the countryside, during which time they tell one another bawdy, silly, comic stories. As Morris points out, this may well be regarded as just as much remedy as pastime, with humour helping to allay the onset of the plague. It is well worth noting that the Decameron is prefaced by a deeply disturbing and descriptive account of the effects of the Black Death (this seems to be a decent source of the text here, if you're interested). The juxtaposition of physical malaise, great social loss, grief and comedy is very explicit here, marking humour as a clear coping strategy in a time of great trial.
I have rarely found on-the-nose jokes specifically about the black plague (though again, I would be delighted to hear the details of any known to exist - seriously, I live for finds like this!) but it's certainly not treated as off-limits for comedy both during, and in the immediate aftermath of enormous loss.