Is information lost when it falls into a black hole?

Perhaps. Whatever structure possessed by any matter that passes over the event horizon is totally lost: the hole is characterized entirely by its total mass, angular momentun and electric charge. This represents a break in causal sequencing which is uncommon in physics.

The problem is compounded by the ultimate fate of a black hole: total evaporation by Hawking radiation.

One way of looking at wavefunction collpsse is to view it as decoherence due to entanglement with the environment - when one adds in such entanglement then one tends to find that many aspects of the quantum superposition vanish naturally, leaving a more classical-like world.

This analysis is frought with subtleties, but for here i will only mention that the whole argument has been criticized by many for mistaking an approximation ("entanglement with the environment") as a resolution of a question of principle ("the world is a superposition"), and as part of this criticism it is often asserted that one could in principle find observables to peel the environment and the system apart again (restoring the superposition)... Here is the present point: if information is pouring down black holes then each moment of the universe is an unmitigated, irreversible entanglemnt.

John Bell called environment decoherence "FAPP - For All Practical Purposes" to belittle it, but if black holes are involved then in a sense the entanglement is not merely FAPP, but perhaps somehow objective.

One could put it prettily: "one moment does not commute with the next..."

I think it was Stephen Hawking who first publically commented on turning a pure quantum state into a classical mixture using a black hole - he mentioned a pure singlet state where we initially treat one end using a density matrix for reasons of ignorance, whereas if the other end fell into a black hole then we would have an honest to god density matrix in our hands! Something like this intuition underlies Roger Penrose's program to link quantum theory and relativity. (For example, see Penrose's conversation with John Baez.)

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See also: Black Holes, Decoherence and Objective Reduction, part I
Quantum-d, 10/6, S Hameroff and L Crowell

Black Holes, Decoherence and Objective Reduction, part II
Quantum-d, 10/6, L Crowell and A Matacz

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See also "The Nature of Space and Time" by Penrose and Hawking in Scientific American, July 1996!

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Cosmic Censorship
No Hair
Entropy
Evaporation
Weyl Curvature Hypothesis