From Applied Cryptography, by Bruce Schneider (pp. 157-158):

One of the consequences of the second law of thermodynamics is that a certain amount of energy is necessary to represent information. To record a single bit by changing the state of a system requires an amount of energy no less than kT, where T is the absolute temperature of the system and k is the Boltzman constant. (Stick with me; the physics lesson is almost over.)

Given that k = 1.38×10^-16 erg/°Kelvin, and that the ambient temperature of the universe is 3.2°Kelvin, an ideal computer running at 3.2°K would consume 4.4×10^-16 ergs every time it set or cleared a bit. To run a computer any colder than the cosmic background radiation would require extra energy to run a heat pump.

Now, the annual energy output of our sun is about 1.21×10^41 ergs. This is enough to power about 2.7×10^56 single bit changes on our ideal computer; enough state changes to put a 187-bit counter through all its values. If we built a Dyson sphere around the sun and captured all its energy for 32 years, without any loss, we could power a computer to count up to 2^192. Of course, it wouldn't have the energy left over to perform any useful calculations with this counter.

But that's just one star, and a measly one at that. A typical supernova releases something like 10^51 ergs. (About a hundred times as much energy would be released in the form of neutrinos, but let them go for now.) If all of this energy could be channeled into a single orgy of computation, a 219-bit counter could be cycled through all of its states.

These numbers have nothing to do with the technology of the devices; they are the maximums that thermodynamics will allow. And they strongly imply that brute-force attacks against 256-bit keys will be infeasible until computers are built from something other than matter and occupy something other than space.

Essentially, the laws of physics prevent the existence of classical computers which can count to 2^256, let alone break 256-bit encryption. Moore's law alone won't ever stop Bitcoin's proof of work from working.

There are two ways around this, however. The first is a break in SHA-256. This is quite unlikely, as Bitcoin uses twice as many rounds of SHA-256 as the usual implementation, as block headers are actually hashed twice. If SHA-256 is ever broken, Bitcoin will actually probably be one of the last programs to suffer from the break, and will have ample time to switch algorithms.

The second way to get around the aforementioned laws of physics is with a computer not made of bits; a quantum computer. Any function can have its domain searched in O[N^(1/2)] on a quantum computer by Grover's algorithm. For SHA-256, with a domain and codomain size of N = 2^256, a string of zero bits twice as long as that of a classical computer running at the same speed can be found in the same time. This would still not be enough to exhaust the remaining bits, and in any case, an attacker with a quantum computer would make much more money by cracking Bitcoin public keys and stealing their balance.

EDIT: Fixed exponents getting mangled in the copy-pasting from PDF.