As a short summary, after a good bit of investigation, I haven't seen any evidence that suggests Lewon II was poisoned.
Nomenclature
Firstly, to be clear in nomenclature, English historiography uses both Levon and Lewon, with Lewon favoured more recently, while he is also known as both Lewon II and Lewon III. Stewart's 'The Assassination of King Het'um II' notes this as "Lewon III is often mis-numbered “IV”." although it doesn't otherwise bear on this specific question (despite being a very interesting article). In any case, when reading about these kings it seems to be quite useful to fix them in time as well as the regnal number to be certain that we are talking about the same people.
Lewon II's Death
The only academic source I found that specifically mentions a cause of death for Lewon II is Boase's 'The Cilician Kingdom of Armenia' which writes:
In 1290, at the age of fifty-three, Leon died worn out by his problems. He had received much affection from his people. Stories of his captivity and his constancy to the Christian faith were made into ballads. 'He was a prince inclined by nature to piety, resolute in adversity and without pride in prosperity. He followed the will of the Lord.'
It is relevant here that Stewart's more recent 'The Armenian Kingdom and the Mamluks' cites Boase regarding the death of Lewon II:
Worn out, Lewon died, aged fifty-three, in 1289.
This is notable because this review of Stewart's work noted that the author rectified a long-standing problem where historians were ignoring Arab sources in treating Cilician Armenia. That Lewon's death stood unchanged through this repositioning of the historical narrative suggests that the Arab histories did not contest, if they mentioned, the death of Lewon II. Boase's sources for Lewon II's rule were:
Vahram of Edessa wrote his rhymed chronicle of the king of Armenia at the request of Leon III, a ruler for whom he had the greatest admiration, and ends it with him still on the throne c. 1280.
The fourteenth century brings us to the important if somewhat enigmatic figure of Hetoum the historian (see supra, pp. 28-30). The Chronology (RHC.Or. I, pp. 488-90) is assigned by a recent editor (V. A. Hakopian) to Hetoum II. For a discussion of the life of the historian see Ch. Schefer in RHC.Arm. II, pp. xxiii-lxxxiv.
Another narrative, Bedoukian's 'Coinage of Cilician Armenia', says the below without a single notion of unfair play:
Levon died on February 6, 1289. Of the eleven children born to him, nine were living at the time of his death. His five sons fought among each other and at one time or other each managed to occupy the throne. Of the two daughters, Zabel married the brother of the king of Cyprus and Rita became the empress of Byzantium. The other two daughters married crusader princes.
Potential Other Narratives
Meanwhile, Shnorhokian has a fairly critical overview of Hayton, or as-mentioned-above "the historian Hetoum", in 'Hayton of Korykos and La Flor des Estoires: Cilician Armenian Mediation in Crusader-Mongol Politics, c.1250-1350'. In this, the author explicitly mentions that the chronicler is hostile against Het'oum II because that king was unwilling to take the throne upon the death of his father, Lewon II. This seems to suggest a lack of motive for any suspicious dealings at least by the eldest son—whom the nobility favoured to take the throne (based on the comments criticising the same chronicler). It sounds that even rumours of suspicious activity might have been reported by Hetoum the chronicler in such an environment.
At the same time, Boase's history also provides a narrative for Occidental characterisation of the Armenians as duplicitious where a rumour of a poisoning, intended or inadvertent, may have originated (though I've been unable to actually find notice of such a rumour):
Such doings hardly served to create a favourable image of Armenia in the West. In the Directorium ad passagium faciendum, a pamphlet addressed to Philip VI in 1332 by a Dominican who had visited Cilicia, the writer warns the king of Armenian unreliability. When hard pressed by the Turks they appeal to Rome but 'the leopard cannot change his spots, nor the Ethiopian his skin: they partake of every error known in the East. . . . Their king (Leon III) had nine children, and all, sons and daughters alike, have come to a violent end, except one daughter and no one knows what her end will be. One brother killed another with the sword; another poisoned his brother; another strangled his brother in prison, so that they all murdered one another till only the last was left and he was poisoned and died miserably.' The truth was not so black, but the family feuds were bloody enough to make it a not surprising picture.
Other Sources
