In most cases, no. There are very few instances where it's used, "He's got the 'flu, the pox, the measles ... are some exceptions." Note that it's usually colloquial or slang terms that take the article.
It's also used when a specific strain of a disease arises somewhere - the Spanish flu, the Asian flu.
But normally, it's "He's got cancer, diabetes, subacute sclerosing panencephalitis, etc."
Note, "the 'flu", but not "the influenza".