I think that the following fields do not care about the age of the Earth a lot:
- Conservation of momentum/Newton's laws
- Some parts of EM (electrostatics)
The following fields will fall apart:
- Maxwell's equations (since these predict the light from stars is long ago)
- Quantum mechanics (everything goes)
- Relativity (all wrong)
The way to find the cutoff is to see when physics started to date the Earth reliably--- this happened around the turn of the 20th century. So all physics before that wasn't very sensitive to the age of the Earth, and all physics after that was. It's not much more subtle than that.
In geology, the evidence for an old-Earth was found in the 19th century, same in biology. So more of those sciences fall apart, essentially all of geology and all of biology.
But physics is a bottom-up science--- the most foundational things are the ones on which everything else is built. So the young Earth, although it is wrecking "only" the last hundred years of physics, is wrecking the most foundational parts, and the parts that have withstood the most scrutiny, and which lie beneath and explain the parts that do happen to work under young Earth.
But one must remember that the adherents of young Earth basically only have a manifestly fictional book to point to, so their evidence has zero credibility. They don't believe this because they have evaluated evidence in a neutral manner.
If you wish to get rid of young Earth creationism, it is important to both respect this belief--- understand its source, and attack the beliefs at the source. Young Earth creationists don't care about Big Bang nucleosynthesis. They care about loving your fellow man. In order to love one's fellow man, one must have a system of absolute ethics, which is roughly along the lines of Hofstadter's superrationality. But extending Hofstadter's concept to asymmetric games is logically-positivistically equivalent to postulating that there is a super-smart agent which knows the right behavior in all circumstances (see this Christianity answer).
Most of these people are not completely delusional, they are only following (understandable and compelling) spiritual guidance that tells them that there is a super-smart agent in charge of ethics, that it is extremely important to inform people of this agent, and that if you have to, you lie through your teeth to get them to believe in this agent, because this trumps all other concerns. I am sure that most of these folks don't actually believe the Earth is 6000 years old, or that there was a global flood, but they are shrewd enough to know that it would be good if people followed the absolute morality, and they are sure that the only path to this absolute morality is found in the Bible.
So they lie about almost anything, including the content of the Biblical texts. This makes them not science-friendly, since science demands honesty as the first and only commandment.
These lies are not tenable in the modern media environment, but it is important in my opinion to preserve the valuable ethical lessons of the Biblical texts, and extract the logical positive essence which can be used to continue religion outside the realm of magical thinking, before cosigning the magical fantasies to the dustbin of history.