No. He died in his sleep. Not from and overdose or something like that. However, he was taking opium for his pain.

Here's a report on his death from the time:

His death is described in the book The Life of Benjamin Franklin. This is the account of Dr. John Jones:

"...when the pain and difficulty of breathing entirely left him, and his family were flattering themselves with the hopes of his recovery, when an imposthume, which had formed itself in his lungs, suddenly burst, and discharged a quantity of matter, which he continued to throw up while he had power; but, as that failed, the organs of respiration became gradually oppressed; a calm, lethargic state succeeded; and on the 17th instant (April, 1790), about eleven o'clock at night, he quietly expired, closing a long and useful life of eighty-four years and three months."

He mentions malaria, then known as the Ague, in his play Henry the Fourth .

"Bootless home and in foul weather too. How 'scapes he ague i' the Devils name?"

He mentions syphilis, often called the French disease or the Neapolitan bone-ache, even more often, but often indirectly. It's one of a bunch of diseases mentioned by the foul-mouthed Thersites in Troilus and Cressida:

Why, his masculine whore. Now, the rotten diseases

of the south, the guts-griping, ruptures, catarrhs,

loads o' gravel i' the back, lethargies, cold

palsies, raw eyes, dirt-rotten livers, wheezing

lungs, bladders full of imposthume, sciaticas,

limekilns i' the palm, incurable bone-ache, and the

rivelled fee-simple of the tetter, take and take

again such preposterous discoveries!