I think that some details would depend on how absolute zero is defined, but in any case, it might be theoretically possible to observe that a system was at absolute zero, but the act of observing it would cause it to no longer be at absolute zero.
Let's try this with two different definitions.
For the first, we'll use a thermodynamic (macroscopic as opposed to microscopic) definition where the temperature is a function of the average kinetic energy of all molecules in a system, and temperature defines the direction of heat flow. Let's pretend that we do have a way of siphoning off all heat to a region of less than absolute zero (obviously not possible in reality, so we'll use a magic freeze-ray) and the result is a collection of molecules with zero kinetic energy.
Now the question is - it has zero kinetic energy relative to what? If we measure velocities of the molecules relative to each other, we can say they have zero kinetic energy. But what about relative to us, the observer? Or to the container that the molecules are in? Since the molecules that we are made of are not at absolute zero, then relative to us, there is still some temperature.
So we use our magic freeze ray on ourselves and the rest of the universe until there is no kinetic energy left at all. Now, we want to observe or measure that we are at absolute zero. We are going to need to come up with some way of measuring the momentum of another molecule relative to ourselves. The least intrusive way of doing this is with light, but even with light, we will introduce some change to the system that will cause it to no longer be at absolute zero.
Let's look at another definition (the one used by @Count Iblis). Here we say that a system in its lowest quantum mechanical energy state is at absolute zero. It's another way of saying that no more energy (of any kind) can be removed from the system. As he said, if you could put such a system in its lowest energy state, then you must have had some way of knowing the current state, and thus, would know it is at absolute zero. This is sort of like the freeze-ray argument from above. However, if we want to observe it at absolute zero (which I think is what you are asking), we need to interact with it in some way. Again light is the best option, but even light will impart some energy to the system, and knock it out of the ground state.
And so in summary, I think we can say that if it were possible to get a system to absolute zero, then you would be able to say it was at absolute zero, but you would not be able to say that it is at absolute zero - you would not be able to observe it without disturbing it.