Word for a task which is flawed or doomed to failure but which you have to do anyway?

Consider Sisyphean:

ADJECTIVE

(Of a task) such that it can never be completed.

EXAMPLE SENTENCES
It was a Sisyphean task - but Paul did not give in.
Not only do they lose the game but they are sentenced to a Sisyphean task.
It was a Sisyphean task of epic proportions that defied a normal life expectancy.

The adjective originally relates to the story of Sisyphus in Greek mythology:

The son of Aeolus, punished in Hades for his misdeeds in life by being condemned to the eternal task of rolling a large stone to the top of a hill, from which it always rolled down again.

The term "death march" was used by Edward Yourdon to describe such futile projects specifically in the workplace. The workers know the project will end in failure, but they are forced to continue anyways.

It's very close to "Dilbertian", in that it makes a cynical yet humorous jab at workplace issues, but it has the same problem of only being known to a narrow audience of people. A more common workplace term is "sinking ship". It can be applied to anything that is on its way to failure while the people involved can only stand on board and watch helplessly. It might also imply that these people want to "jump ship", i.e. change jobs.

I'm surprised by the absence of fool's errand. Perhaps appending "obligatory" or "mandated" to the front to meet the requirements of the question.

Apparently, it means snipe hunt which could be another one or you could click through all the links you find which would be a fool's errand of its own.