Is a dice roll deterministic? This should be understood as distinct from the question of is it possible to predict the outcome of a roll, which seems to be an issue related to intractability and observation?
What I'm really asking is, if quantum indeterminacy is a factor of more than just observational limitations and intractability, where outcomes may be independent of of the prior state of the system, does it affect outcomes in the world of classical mechanics?
My interest is related to certain combinatorial problems, whether random number generation needs to be integrated, and if so, how it might be treated differently from uncertainty arising out of imperfect or incomplete information or intractability. 
 A: Quantum effects will be too small to effect a macroscopic objects like dice. From classical mechanics if the initial conditions are the same then the final condition will be the same. Of course one cant get the initial condition exactly the same, but it can be made close enough for a coin toss so probably could for dice as well.
A: 
What I'm really asking is, if quantum indeterminacy is a factor of more than just observational limitations and intractability, 

Indeterminancy is inherent in the probabilistic postulates of quantum mechanics, where only the  probability of getting a single measurement can be calculated exactly, and not the number.

where outcomes may be independent of of the prior state of the system, does it affect outcomes in the world of classical mechanics?

Let us take a perfect dice: the probability distribution will be flat at 1/6 for any throw.
Suppose you get from a specific dice this plot:

Your question amounts to asking :can this bias be due to quantum mechanical effects?
The general answer is that quantum mechanics describes dimensions commensurate with h_bar and the number of molecules in a dice are of order 10^23 and the statistics will decohere a usual ensemble of atoms . BUT cystals are a macroscopic , dimensions of a dice, manifestation of quantum mechanics, as is crystal growth. Thus I could think of a way of biasing  a dice using quantum mechanical knowledge: for example build one face of the crystal with a heavier isotope.
So the answer is very improbable unless extra measures are taken. 
PS. Maybe this answer of mine for a different question might interest you.
A: You are asking about the origin of probability in dice rolls. As argued convincingly by e.g. Jaynes in LoS, probability in dice rolls and coin tosses originates from our ignorance of the initial conditions. Were we cognisant of the initial state (the position and velocity of the coin or dice) with sufficient precision, we could evolve them in time and determine the final state. The fact that we might suppose every outcome of dice to have equal probability is a reflection of our ignorance about the dice (it may have imperfections) and the mechanism by which it is thrown (which may favour particular outcomes). Even if it were the case that the behaviour of the dice was quantum mechanical or chaotic, our choice that $p=1/6$ would still represent our limited knowledge of the dice-thrower system.
