Does the uncertainty principle make simulation of systems impossible? Is it possible to fully define a system, then be incapable of simulating or calculating its future states due to the Uncertainty Principle? If it can be done, how?
 A: The Uncertainty Principle will never, as far as we know, prevent you from simulating any physical system. The reason for this is that quantum mechanics is - except for that little problem with measurements - completely deterministic.
To be more precise, say you want to simulate a given system within quantum mechanics. You begin by describing your preparation procedure of the initial state, you describe the hamiltonian which drives the evolution of the system, and you describe any measurements you will do at any given point. Then quantum mechanics allows you to calculate, at least in principle, the evolution of the system's state via the quantum Liouville equation,
$$i\hbar\frac{\partial\rho}{\partial t}=[H,\rho].$$
When you perform measurements, the formalism will tell you the probabilities of each outcome and the state you should use to continue the unitary evolution. The whole thing is completely simulatable. (On the other hand, there is no guarantee on you being able to find a computer that will do this in less than the age of the universe.)
Even in classical mechanics, this is not an issue. Say you have a classical particle which you want to simulate using some hamiltonian mechanics, but you're worried that you can never have full information about both position and momentum. The Uncertainty Principle does limit your precision to a patch of area $\hbar$ in phase space. However, your preparation procedure will produce some sort of definite probability distribution over phase space which determines what positions and momenta are more likely than others. This probability density can then be propagated deterministically in time using liouvillian mechanics. This formalism will give you, at any given time, the probability distribution over the position and momentum of the particle; if you repeat the experiment over your ensemble then you can simulate the distribution of final values.
A: As Emilio pointed out, the uncertainty principle is not a limiting factor. However, as for simulating or calculating future states, this is not really generally possible for classical systems, because of chaotic behaviour. 
A: Let's start off by removing the restriction of computational resources such that we're not limited by computing power and by finite precisions. Let's also use the word exact to mean absolute certainty (ie. probability is precisely 1) about a quantity. 
Take a real group of particles at an initial state. We may or may not be able to derive a set of governing equations that is completely deterministic in the sense that given the exact initial momentum and exact initial position of each particle in the system, we could solve for the exact momentum and position at any other time. But even if such a governing equation existed, we could never take a real system and translate it into initial conditions because we can never measure the exact momentum and exact position to start the simulation. We are limited by the Uncertainty Principle, so no matter how deterministic the equations may be, we cannot be more precise than our uncertainty in the initial conditions.
Now, we can change variable such that we're using a wavefunction or probability density as our variables. These may be solved deterministically, and we may know them exactly from our real system to use as initial conditions. That's awesome, but it means all we can do in the limit of infinite resources is come up with the exact probability of the state of the system at a future time.
But we cannot turn those probabilities into an exact momentum and exact position, no matter how exact the probabilities are. We can be certain to any reasonable amount that our particle is at a position and momentum, but we can never be exactly certain. 
It's being pedantic of course, but much of the limiting cases tend to be. In a practical world, we can get good enough or better than we could ever manufacture real equipment to measure. But in theory, it is fundamentally impossible to simulate exact momentums and exact positions of real systems because we could never know the initial state exactly.
