How did the Emperor Titus die?

He could have had an advanced vestibular schwanoma which can both cause tinnitus and press against the hypothalmus resulting in an elevated body temperature.

SHORT ANSWER

The most widely accepted view among scholars is that Titus died a natural death. He had a fever which may have been caused by malaria (note that its relationship to mosquitos was unknown at the time) or (taking longer term symptoms into consideration) possibly a brain tumor. The possibility that his brother Domitian may have hastened his end cannot be ruled out.

DETAILED ANSWER

Suetonius stated he died of a fever. Cassius Dio says that ‘some writers’ say that Titus died a natural death but also mentions that

The common report is that he was put out of the way by his brother, for Domitian had previously plotted against him; .......The tradition is that, while he was still breathing and possibly had a chance of recovery Domitian, in order to hasten his end, placed him in a chest packed with a quantity of snow, pretending that the disease required, perhaps, that a chill be administered. At any rate, he rode off to Rome while Titus was still alive, entered the camp, and received the title and authority of emperor,

In other words, this tradition has it that Titus was helped on his way by his younger brother Domitian. Illustrated History of the Roman Empire mentions another story with Domitian as the villain:

Some rumours claim the emperor's death was not at all natural, but that he was killed by his younger brother Domitian with poisoned fish.

That the relationship between Titus and Domitian was far from one of brotherly love cannot be disputed, but there is the age-old problem of biased sources, in this case very much to the detriment of Domitian. Brian Jones, in The Emperor Domitian (1993), says of Domitian:

even though (so he claimed) he was left a share in imperial power according to the terms of his father’s will, he still received nothing, since the will had been tampered with — and Titus was known as an expert forger. That Domitian was dissatisfied with his lot is not unlikely, but the extent of his reaction is hard to assess, given the bias of our sources

Jones also refers to Domitian’s alleged plots against Titus but, nonetheless, he stops short of concluding that Domitian killed Titus, instead noting that

whatever the relationship between them, Domitian seems to have displayed minimum concern for Titus during his illness in September 81. As the emperor lay dying on the 13th, Domitian made for the praetorians’ camp, promised them a donative and was hailed as emperor

Christopher Scarre, in Chronicle of the Roman Emperors (2012), agrees with Jones. On the subject of the stories about Domitian being implicated in Titus’ death, he says:

All this, however, is part of the later attempt to blacken Domitian's reputation in every way possible, and there is no good reason to suspect that Domitian acted improperly.

That Titus’ fever might have stemmed from malaria is mentioned by Scarre and also by Rachel Rechthand in The Gnat that Killed Titus.

Based on the text of the Talmud, S. J. Bastomsky noted that the word used for ‘gnat’ in the Talmud, yattush, could also be translated as ‘mosquito’, which might suggest that Titus died of malignant malaria.

However, although malaria was widely recognized in ancient Greece and Rome, Robert Sallares in Malaria and Rome (2002) says that

In Europe the theory that mosquito bites caused malaria was first proposed in print by Giovanni Maria Lancisi....published in 1717

so S. J. Bastomsky's use of the Talmud is on shaky ground without any evidence of an early knowledge of the connection between malaria and mosquitoes. Miasma theory was the commonly accepted reason for malaria in ancient times. This does not mean that Titus didn't die of malaria, just that Bastomsky seems to have misused evidence (unless he can prove that the Talmud writers knew of the connection between malaria and mosquitoes).

Another theory is that Titus died of brain cancer. Rechthand says that:

Roman accounts of the events leading up to the death of Titus fit in with the theory that Titus died of a brain tumour. Suetonius and Dio...speak of what could be described as depression

Referring to Cassius Dio, Rechthand says that after the late summer of 80 C.E., Titus achieved ‘nothing further of importance’. She concludes:

So it seems that right before his death, Titus suffered from a period of depression, in which he accomplished little, and then died....There are very few modern cases of individuals suffering from brain tumours who managed to carry on with their lives at a routine level.

Although it is highly unlikely that we will ever know the true cause of Titus’ death, it would seem that, on balance, it was most likely a natural one. We should beware of taking ancient sources too literally. The Jewish Encyclopedia states that almost all the accounts of Titus in rabbinical literature are ‘purely legendary’. At the same time, Roman sources are heavily biased against Domitian, though we cannot totally discount the possibility that Titus’ younger brother helped him into his grave a little quicker than might otherwise have been the case.

Other sources:

The Oxford Dictionary of the Classical World

Domitian's Rise to Power by Kathleen D Toohey

A History Of The Roman World From 30 BC To AD 138 by E. T. Salmon