Given his age of 72, anything associated with 'old age' would probably serve as a cause of death. Apparently, he suffered with alcoholism, but this first notation is the closest I found to an actual cause:
The rent was paid by his friend Walter Morton, who also paid for house calls by Dr. James Manley, a Christian physician who kept his beliefs mostly to himself. One biographer suggested Paine had arteriosclerosis of the brain.
Derided by the public and abandoned by his friends, he died in 1809 in New York City, a drunk and a pauper. The whereabouts of his remains are unknown today.
http://www.ushistory.org/paine/ Hooted upon the streets, lampooned in the newspapers, deserted by his political associates, he lived a wretched existence. He was buried on his farm in New Rochelle, but his remains were removed to England in 1819 by William Cobbett. What became of them is unknown.
https://en.wikipedia.org/wiki/Thomas_Paine#Death Increasingly neglected and ostracized, Paine's last years were marked by poverty, poor health and alcoholism. When he died in New York on June 8, 1809, he was virtually an outcast. Since he could not be buried in consecrated ground, he was laid to rest in a corner of his small farm in New Rochelle.
http://www.historyguide.org/intellect/paine.html