More than one source describes how Mehmed had been ill during the preceding winter though his death was "unexpected"—implying he had healed from the winter illness. Nevertheless, Mehmed fell ill on May 1st and died on May 3rd, a few days later. The account of his poisoning is described as "circumstancial" though also "substantial" in that encyclopedia while most authors exhaust the various possibilities to end up at the "We don't know" point-of-view.
In the last days of April 1481, Sultan Mehmed crossed the Bosporus to the army mustering-ground at Üsküdar, ready to lead his army through Anatolia. On 3 May, only one stage further on, at a spot near Maltepe known as ‘Sultan’s Meadow’, he died, aged 49, possibly from complications associated with his gout.80 Although he had a history of ill-health his death was unexpected, and he had not designated a successor.
80: Vatin and Veinstein, ‘La mort de Mehmed II’ 187–90
—Finkel, 'Osman's Dream'
Finkel's is, however, a very broad overview of the Ottomans. A more specific one is given in Babinger's 'Mehmed the Conqueror and His Time' (all bold emphasis mine):
Mehmed, however, was not well. He spent the winter quietly at his serai in Istanbul. ...
On April 25, a Wednesday, Mehmed crossed over to Üsküdar and the march began. A halt was made on Emperor's Meadow (Hunkar Cayiri) near Gebze, not far from the place where Hannibal was buried. Here, on May 1, the sultan was stricken with intense abdominal pains and the physicians were summoned. In addition to his old ailments, gout and rheumatism, new ones had set in. The first of the sultan's physicians to try his hand was the Persian Hamiduddin of Laristan, hence surnamed al-Lari. His role during the last days of the Conqueror's life was more than suspect. In fact, he had so little success in dispelling the widespread doubts concerning his trustworthiness that when he died four years later (February 22, 1485) in Edirne, where he had taken up residence and begun to build a mosque (completed in 1514), local legend attributed his death to an overdose of opium forced on him by Bayezid II. The version most favorable to al-Lari is that he merely gave the sultan an inappropriate medicine by mistake. Thus, it must be supposed either that the purpose of doing away with al-Lari was to eliminate an undesirable witness to a possible attempt to murder the sultan or that he was actually responsible for Mehmed's death.
When al-Lari failed, the sultan's old confidant, Maestro Iacopo, was called to the sultan's sickbed. He declared, however, that there was nothing he could do, because his predecessor had used an unsuitable medicine, the effects of which he could no longer combat. The pain became excruciating; the potion administered to the dying sultan seems to have blocked his intestines. At the time of the afternoon prayer, or about four o'clock, on Thursday, May 3, 1481, Mehmed the Conqueror, then aged 49, gave up the ghost. Mars governed the hour, as the Ottoman chroniclers expressly note.
We cannot be absolutely sure about the cause of Mehmed's death. The large number of his enemies and certain circumstances attending his death make it seem likely that he was poisoned. ... it must be assumed that the state of his health was satisfactory when he left his capital on April 25, especially in view of eyewitness reports that the fatal intestinal pains set in suddenly on the following Tuesday. These considerations favor the hypothesis that he was poisoned immediately after his departure and that no medicine could have saved his life. If he was poisoned, we do not know who was behind it. In view of the fact that a dozen previous attempts by the Venetians to murder the sultan had been lamentable failures, it seems relatively certain that the Venetians had nothing to do with it. The sultan's son Bayezid seems a far more plausible candidate. The relations between the freethinking father and the mystical, bigoted son had never been cordial.
Bayezid's potential motive is described as related to Mehmed I and the grand vizier, Karamani Mehmed Pasha, wanting to raise Prince Cem as future sultan in Bayezid's place:
This view is supported by the pointed words of Muhyiddin Mehmed, nephew of the great astrologer Ali Kusci and sheikh of the Halveti, who died in the odor of sanctity in 1514 in Iskilip (northeastern Anatolia). Shortly before the Conqueror's death, he went on a pilgrimage to Mecca; in Amasya he took his leave of Bayezid (who later as sultan accorded him very special favor) with the significant words that by the time he returned, the prince would have become sovereign in his own right. The mysterious remark was later interpreted, according to the custom of the dervishes, as proof of the sheikh's gift of divination and hence sanctity. As for Bayezid, he more than once expressed the opinion formulated in the Arabic saying "There are no ties of kinship between princes" (La arhama baina 'l-muluk). Thirty-one years later he himself was to fall a victim to the same harsh code: he died on May 26, 1512, on a trip to his native Dimotika, of a poison administered to him by a Jewish physician at the behest of his son and successor, Selim. In any event, it is safe to say that while Mehmed the Conqueror may have been suffering from the gout, he was not fatally ill when he set out on a campaign which, to judge by the scope of the preparations that had been made, must have been a large-scale undertaking.20
20: On the sultan's death see the article by Babinger cited above, book 2, n. l 5. A Turkish translation, with additional material, was published by Feridun N. Uzluk, Fatih Sultan Mehmed zehirlendi mi eceli mi öldü? (Ankara, 1965). Strongly critical of Babinger's verdict of poisoning in the death of the sultan is M. C. Sehabeddin
Tekindag, "Fatih'in ölümü meselesi," Tarih Dergisi 16 (1966), 95-108. Cf. also the remarks of Kissling concerning the involvement of the Halveti dervishes (see above, book 4, n. 37) . And cf. Tahsin Yazici, "Fetih'ten sonra Istanbul'da ilk halveti seyhleri: Celebi Muhammed Cemaleddin, Sünbül Sinan ve Merkez Efendi," lED 2 ( l 956), 87-1l3.