What do you call a person who is chasing a hopeless or lost cause?
I'd call them Quixotic [Oxford]
Extremely idealistic; unrealistic and impractical:
However, Quixotic usually describes ideas (as against the three words in its definition, which can describe both ideas and people). To describe people, I think you might have to go with the character name: Quixote
EDIT: Based on the comments, I think most people, like me, are okay with Quixotic in this sense. And as @Oldbag points out, we also have 'Quixotical' (though he doesn't think Quixotic is wrong)
A term gaining traction is the incurable optimist [Forbes]
If you're flexible on the one-word restriction, you could say:
He's hoping against hope.
a person who is chasing a hopeless or lost cause?
There is set phrase that actually describes exactly your person and it is: