What illness did Tanjiro's dad have?
Its never stated in the manga why he is so sick.

But if you read the manga you will have a theory


  Tanjiro's dad have The mark, Its stated by upper moon 1 every demon slayer who has the marked will be died before they turn 25 years old. We don't know if Tanjiro's dad have the mark or even he is a demon slayer before, but because he can use breathing technique Dance of the Fire God it's safe to assume he was once a demon slayer. even though it doesn't guarantee if tanjiro's dad ever have The Mark, because not every Demon Slayer have The Mark.