A word for an inescapable and pointless situation?

"Quagmire" is certainly a fitting word for your example:

"The American government has decided to pour extra money into the war in Iraq, unable to escape the quagmire it has entered."

Definition from M-WO:

2: a situation that is hard to deal with or get out of : a situation that is full of problems

The first definition (also from M-WO) is helpful in understanding how "quagmire" came to be used for other tricky, dangerous situations:

1: soft miry land that shakes or yields under the foot

EDIT: "Quagmire" was apparently quite commonly used to describe the US' involvement in the Vietnam War as is illustrated in the book, "The Making of a Quagmire: America and Vietnam During the Kennedy Era" by David Halberstam, Daniel J. Singal (Editor & Contributor)

The answer to the title is kafkaesque.

of, relating to, or suggestive of Franz Kafka or his writings; especially : having a nightmarishly complex, bizarre, or illogical quality <Kafkaesque bureaucratic delays>

But the answer in your example is quagmire.

In Greek mythology Sisyphus was punished with the task of rolling a boulder uphill for eternity. Each day the boulder would roll back down and he would have to roll it up again. This endless, pointless toil lends us

Sisyphean Endless or unavailing (Dictionary.com)

and thus giving you a sisyphean task

rathole

US a bottomless pit. (Typically with throw and down as in the examples.) Why do they keep throwing money down that rathole?

McGraw-Hill's Dictionary of American Slang and Colloquial Expressionsp

"If you want to see a change, the only way that's going to happen is if we elect Joe and others like Joe to join with the increasingly large number of Republicans who have seen the light," he continued. "If we elect Joe's opponent and his colleagues, nothing is going to change in Iraq, except we're going to continue to go down this rathole."

Journal Inquirer

The American government has decided to pour extra money into the war in Iraq, unable to escape the rathole it has gone down.

goat rope

also goatfuck (US military slang, vulgar)

A confusing, disorganized situation often attributed to or marked by human error.

Wiktionary

hornet's nest

A highly contentious or hazardous situation.

WordNet by Farlex

A troublesome situation or place in which there are many dangers.

M-W

What I was talking about was this administration’s decision to make Iraq part of its war on terrorism, but what I was thinking about was Vietnam. It was not because of nostalgia or political partisanship (after all, it was liberal Democrats who got us into the war in Southeast Asia and neocon Republicans who sent us into the hornet’s nest of Iraq) that I thought about and mentioned Vietnam, but because it was the same kind of dangerous spin that had given us Vietnam and dragged us into Iraq...

Huffington Post

While the president struggled to keep his promise of withdrawing from the hornet's nest of Vietnam, Congress showed its newfound power by adopting a series of amendments restricting the ability of the U.S. to go to war...

Let My People Go

The American government has decided to pour extra money into the war in Iraq, unable to escape the hornet's nest it has entered.