Venice, C14th

"Corpi morti" ("dead bodies") was simply an equivalent of "bring out your dead": this is what would be shouted to the occupants of houses as the boats waited or passed outside. The cry is mentioned in John Julius Norwich's classic A History of Venice (1982), and repeated by Thomas Madden in his recent Venice: A New History (2012). (I've checked a couple of other books on Venice (Lane, Ferraro, Queller) and a few papers, which don't specifically mention disposal of bodies during the Black Death - although some address later customs during the early modern period of endemic plague.)

Madden adds that due to the number of dead, boats larger than gondolas (roundships and barges) were also employed to take the boats to uninhabited islands in the lagoon, to bury them in mass graves. (The bodies were not being thrown in the sea.)
Neither Norwich nor Madden references a source. There is nothing extraordinary about the call or the collection of bodies, as it mirrors well-known customs with carts in land-based cities - but I am also hitting the language wall here, as sources which might trace this account specifically to the Black Death of the mid 14th-century, or to some other time, will be in Italian and Latin.

I'm not thrilled to be quoting as old a book as Philip Ziegler's Black Death, but he does at least source his paragraph about Venice's treatment of the bodies.

At the worst of the Black Death six hundred Venetians a day were said to be dying; this rate can hardly have been sustained for long but is by no means incredible.24 On 20 March, 1348, the Doge, Andrea Dandolo, and the Great Council appointed a panel of three noblemen to consider measures to check the spread of the plague.25 A few days later the panel reported its recommendations. Remote burial places were designated; one at S. Erasmo at what is now the Lido, another on an island called S. Marco Boccacalme which seems since to have vanished into the lagoon.26 A special service of barges was provided to carry corpses to the new graveyards. All the dead, it was ruled, were to be buried at least five feet underground.

24 Chronicon Estense, Muratori, 15, III, p.162. [14th century chronicle from Ferrara]
25 Lorenzo de Monaci, Chronicon de rebus Venetorum, Brunetti, ‘Venezia durante la peste’, Ateneo Veneto, 32, 1909. [de' Monaci was a chronicler thought to have been born in Venice in 1351, as the Black Death was subsiding]
26 d’Irsay, ‘Defence Reactions During the Black Death’, Annals of Medical History, IX, 1927, p.171.

The five feet burial depth is also mentioned by Norwich.

There are a number of publications about medical customs in early modern Venice and Italy, and it's widely established that during that period certain locations were used for quarantine purposes. This practice began after the Black Death, at Ragusa (Dubrovnik), a Venetian colony, in 1377, and is known in Venice itself from 1403, where certain islands were reserved for infectious disease patients and people under quarantine. A report of an archaeological excavation of one of these Venetian islands can be seen here

It's quite interesting to note the different characterisations in the historiography. Madden:

Venetians were not as quick to blame the Church or to seek celestial reasons for the Black Death. Perhaps it was the fact that, unlike most Europeans, Venetians traveled to distant lands, where they saw that the plague had devastated the Mongols and the Muslims just as thoroughly as it had the European Christians. Instead, Venetian responses to the plague were much more pragmatic. They prayed, of course, but from a medieval perspective that was the height of pragmatism. They also worked hard to lessen the effects of the plague.

He then goes on to list the above practical measures. Ziegler's scene setting is classically 'medieval', and his mentions of Venice implicate the city in terror of the wrath of god and portents as much as anywhere else.

In a city where much of the transport took place by water, there was nothing extraordinary about transporting dead bodies by boat. However, if Dobson is suggesting they were thrown in the sea en masse, that would be incorrect. (One can't say none ended up in the lagoon or the sea - perhaps because boatmen were fed up or unwell themselves, or there was no space in some pit - but there wasn't a large-scale plan to tip them in.)

Constantinople, C6th

The main primary source for the 542 plague in Constantinople is Procopius, and sure enough, that is where this comes from. History of the Wars, II.xxii-xxxiii:

So those who had not as yet fallen into complete destitution in their domestic affairs attended individually to the burial of those connected with them. But Theodorus, by giving out the emperor's money and by making further expenditures from his own purse, kept burying the bodies which were not cared for. And when it came about that all the tombs which had existed previously were filled with the dead, then they dug up all the places about the city one after the other, laid the dead there, each one as he could, and departed; but later on those who were making these trenches, no longer able to keep up with the number of the dying, mounted the towers of the fortifications in Sycae [Galata], and tearing off the roofs threw the bodies there in complete disorder; and they piled them up just as each one happened to fall, and filled practically all the towers with corpses, and then covered them again with their roofs. As a result of this an evil stench pervaded the city and distressed the inhabitants still more, and especially whenever the wind blew fresh from that quarter.

At that time all the customary rites of burial were overlooked. For the dead were not carried out escorted by a procession in the customary manner, nor were the usual chants sung overthem, but it was sufficient if one carried on his shoulders the body of one of the dead to the parts of the city which bordered on the sea and flung him down; and there the corpses would be thrown upon skiffs in a heap, to be conveyed wherever it might chance.

Specialists in Byzantium will have more to say on the reliability of Procopius. A long time ago I did an essay on the Justinian Plague for a tutor who was a Byzantine specialist, and what I gathered there and elsewhere since was that The Secret History may be treated as scurrilous gossip, whilst the History of the Wars, a more official document, can be regarded much as one would other medieval chronicles.
(Other opinions on Procopius from an old AskHistorians answer from u/GeorgiusFlorentius and a recent review paper by Geoffrey Greatrex, 'Perceptions of Procopius in Recent Scholarship', Histos 8 (2014))

Peter Sarris, in a 2007 chapter 'Bubonic Plague in Byzantium: Evidence from Non-Literary Sources', notes:

John of Ephesus also records Justinian ordering the digging of mass graves at Galata, and further suggests that considerable problems were encountered in disposing of corpses at Jerusalem and the cities of Palestine. John claims even that many corpses in Constantinople were simply thrown into the sea.

With regard to the historiography, Sarris states:

In recent years, however, a marked tendency has emerged to seek to downplay the effects of the late antique bubonic plague, and argues that eyewitness accounts of the disease are hysterical in tone and misleading in effect.43 This revisionist tendency can be dated back to the publication in 1989 of Durliat’s “La peste du VIe si`ecle, pour un nouvel examen des sources byzantines.”44 This article, characterized at the time by Biraben as “une pol´emique constructive,” has greatly influenced subsequent writing on the sixth century.45 Durliat proposed that reliance upon the literary sources had led scholars to exaggerate the impact of the plague on the Byzantine Empire. The testimony of the literary sources for the plague having had a dramatic impact on imperial fortunes and, in particular, on population levels within the empire was, he suggested, gainsaid by the rather more mute and hitherto ignored testimony of the non-literary evidence...

In his opinion, that revisionism went a bit too far:

Durliat arguably had unrealistic expectations of the epigraphic record, given the circumstances in which the inscriptions on which he is dependent would appear to have been made.. In spite of Durliat’s assertion, the claim [by Procopius and others] that the plague occasioned large-scale agrarian depopulation would appear to be strongly corroborated by the numismatic, legal, and papyrological evidence.

in Lester K. Little (ed.) Plague and the End of Antiquity: The Pandemic of 541-570 (Cambridge University Press, 2007).

During the outbreak in Constantinople, whilst there were some attempts to cope in an organised fashion, it appears that it may have simply become overwhelming: two different primary sources suggest that the sea became a means of disposing of dead bodies from the plague.

---

Not sure if anyone else will read this who'd get this, or share my sense of humour, but am more amused than I should be by the title of a paper, THE PLAGUE OF 542: NOT THE BIRTH OF THE CLINIC (John Atkinson, Acta Classica, (2002)) which I found while researching this answer. The caps helped too. At least there's something optimistic in addressing early medieval medicine along those lines.