Yes, all the standard polyhedral dice used in D&D (d4, d6, d8, d10, d12 and d20) are "fair".
In particular, the following assumptions are sufficient to guarantee that a polyhedral die will land on each of its sides with equal probability:
- The shape of the die is isohedral, meaning that for any pair of faces there is an isometric symmetry (i.e. a rotation and/or a reflection) that swaps those faces while leaving the shape of the die unchanged. All of the standard D&D dice, including the d10, are isohedral.
- The die is made of a uniform material, i.e. it is not "loaded" to favor one side. This is generally true of properly (and honestly) made dice, although e.g. some fancy "textured" dice can have a noticeably non-uniform weight distribution.
- The effect of the markings on the faces of the die (which necessarily represent a slight departure from isohedrality and/or uniformity) on its physical behavior when rolled is negligible. In practice, this also seems to be generally true.
- The die is rolled starting from a random orientation, and the person (or mechanism) rolling the die does not look at it before deciding how to roll it.
In particular, assumptions 1–3 essentially guarantee that the physics of rolling the die will not intrinsically favor any side above the others, while assumption 4 says that the initial orientation of the die also doesn't favor any side.
The extra assumption 4 is needed because I have not assumed anything about how the die is rolled — for all we know, it could be just dropped straight down or even simply placed down on the table. In practice, of course, we hope that the bouncing motion of a properly rolled die is itself sufficiently chaotic (i.e. highly sensitive to even tiny variations in initial conditions) as to render the final orientation effectively random even if the initial orientation of the die is not. But it's hard to determine exactly how much shaking and rolling is actually enough to achieve that, so it's easier to just dodge that issue by assuming that the initial orientation of the die is already random and just show that the shape of the die itself won't introduce any bias to the results.
As noted in the MathOverflow thread mentioned by Mark S. in the comments, the assumption of isohedrality isn't strictly necessary; there's a mathematical argument that there must be fair non-isohedral dice too. Well, sort of…
For a gist of the argument, consider taking a thin round coin (with two identical flat sides and a very narrow cylindrical edge) and stretching it out into a long cylindrical rod with the same diameter. Clearly the coin will almost always land on one of its two flat sides and almost never on the edge, while the long cylinder will almost always land on its round side and almost never on one of the flat ends. In particular, as the probability of the coin/cylinder landing on its round edge/side is a continuous function of its thickness/length, there must be some halfway point (looking like a very thick coin or a rather short and stubby cylinder) where the probability of it landing on its edge is exactly equal to that of landing on one of the two flat sides, making the "coin" a fair non-isohedral three-sided die.
However, the problem with this argument is that it only proves that such a halfway point must exist for any particular way of rolling the die — it does not prove that the same halfway point works regardless of how the die is rolled, and indeed there's no reason to expect that it would. So, in effect, the "fair" non-isohedral die constructed in this way will only be fair if rolled with a particular amount of force on a particular surface using a particular rolling motion. Basically, if the die is fair for you, it probably won't be fair for me, and vice versa. Which isn't really very useful or fair at all, when you think about it.
For isohedral dice, on the other hand, the symmetry of the shape forces a corresponding symmetry on the physics of the roll — in effect, it ensures that the sides of the die are indistinguishable as far as the physical laws governing the motion of the rolled die are concerned. Thus, a uniform isohedral die will be fair regardless of how it's rolled.
FWIW, we can relax the isohedrality requirement slightly and still maintain fairness regardless of the rolling method. In particular, it's not quite necessary for all sides of the die to be interchangeable by an isometric symmetry — it's enough that this holds for all sides that the die can land on in a stable orientation, or at least for all sides that are actually counted as valid rolls.
For example, a dreidel is an example of a highly non-isohedral object which, nonetheless, is shaped so that it can only come to rest on a flat surface (with a non-negligible probability) in one of four symmetric orientations, and thus behaves like a fair die. The same is true for other kinds of teetotums and long dice, including the "barrel dice" sometimes used in RPGs.
Also, as noted above, even a simple coin used as a two-sided die technically also has a third side (the edge) that it can land on, which is not isometric with the two flat sides. But the edge is narrow enough that this is very unlikely to happen, and if it did happen, one would generally just flip the coin again until it came to rest one one of the two flat side, which are (approximately) isometric with each other.