Are different interpretations of quantum mechanics empirically distinguishable?

It's not true that all the different interpretations are not (in principle) experimentally distinguishable. Let's consider the difference between Copenhagen Interpretation (CI), De Broglie–Bohm theory (BT) and the Many Worlds Interpretation (MWI). BT assumes that under normal circumstances we have so-called quantum equilibrium and only then do you get the usual predictions of standard quantum mechanics that you get when you assdume CI. This means that you can try to detect small deviations of exact quantum equilibrium, see here for details.

If the MWI is correct then time evolution is always exactly unitary. The CI doesn't explain how we get to a non-unitary collapse, but it does assume that there exists such a thing. This implies that at least in principle there should be detectable effects. Systems that are well isolated from the environment should undergo a non-unitary time evolution at a rate that is faster than can be explained as being caused by decoherence by the residual interactions it still has with the environment.

David Deutsch has proposed a thought experiment to illustrate that MWI is not experimentally equivalent to CI. Suppose an artificially intelligent experimenter is simulated by a quantum computer. It will measure the operator A = |0><0| - |1><1|. The qubit is initialized in the state |1/sqrt(2)[|0> + |1>]. Then the CI predicts that after the measurement the state of the qubit undergoes a non-unitary collapse to one of the two possible eigenstates of A, i.e. |0> or |1>. The MWI asserts that the state of the entire quantum computer splits into two branches corresponding to either of the possible outcomes.

To decide who is right, the experimenter decides to let the computer perform the unitary time evolution corresponding to inverting the final state of the quantum computer (according to the MWI) to the initial state, but while keeping the record that a measurement has been performed. This transform to the modified initial state is still unitary and can therefore be implemented (all unitary transforms can be implemented using only the CNOT and single qubit rotations).

Then it is easy to check that if the CI is correct that you don't get that desired modified initial state back and the difference between the two states if the qubit you end up with, can be easily detected by doing measurements on it.

No, interpretations of quantum mechanics are not distiguishable in a physical experiment, otherwise they would be called theories rather than interpretations.

It should be noted that there are some theories whom their authors call "interpretations" but they in fact are not. For instance the "objective collapse theories" often (wrongly) called "interpretations". These theories can be physically proven or disproven and predict different observations than the standard quantum mechanics (with all its interpretations).

That said, it is not that interpretations cannot be experimentally distinguished at all. Maybe they can be, but the experiment that would be able to distinguish between them would not be a physical (or scientific) experiment in the sense it would not satisfy the requirements for scientific method.

All scientific experiments regarding quantum mechanics should produce the same results as far as different interpretations concerned.

On paper, the difference between a theory and an interpretation is very clear. If two distinct sets of ideas/explanations generate experimentally distinguishable predictions, then they are two different theories. If they use different concepts but produce identical experimentally verifiable predictions, then they are two different interpretations of the same physical theory.

In practice, things are never so clear-cut. As a practical matter, we certainly do not currently possess either the theoretical or the experimental capability to unambiguously falsify either the many-worlds or Copenhagen interpretation in any experiment that can actually be performed in the real world. But as Count Iblis points out, there probably are some hypothetical experiments that distinguish them, which could be performed in principle but may never actually be feasible in practice. (I think the same is true of the de Broglie-Bohm interpretation, although I'm not familiar enough with that interpretation to say for sure.) For example, decoherence is defined to be the vanishing of all the elements of a system's reduced density matrix that are off-diagonal in some basis known as the "pointer basis". But perfect decoherence is an idealization; in practice the off-diagnonal elements will be exponentially small but never exactly zero. (Similarly, there will always be incredibly many-point tunneling terms connect different branches of the wavefunction that appear at zillionth order in perturbation theory.)

So as practical matter, the many-worlds and Copenhagen interpretations are equivalent as physical theories to within our current understanding. In principle, they are almost certainly distinguishable if one could perform arbitrarily precise experiments and had infinite processing power. I would argue that we do not yet understand the different interpretations well enough to judge whether they will ever be experimentally distingishable in practice with foreseeable technology.

So you could approach this situation in many ways, and take any of the following (defensible) perspectives:

1. MWI's and Copenhagen's predictions are indistinguishable in practice, so they are different interpretations of the same theory.
2. MWI's and Copenhagen's predictions are distinguishable in principle, so they are different theories but are equivalent in practice.
3. We don't understand MWI and Copenhagen well enough to judge whether they will be experimentally distinguishable in the future. So we do not yet know whether they are different theories or simply different interpretations.
4. We don't understand MWI and Copenhagen well enough to judge whether they will ever be experimentally distinguishable in the future. But we should reserve the term "theory" to mean "something that we actually know how to distinguish in practice", and use "interpretations" to mean "ideas that we do not currently know how to distinguish experimentally". So the distinction between a theory and an interpretation is dependent on the scientific community's current knowledge of the subject. MWI and Copenhagen are currently just two different interpretations of the same theory, but way well eventually become two different theories as our understanding and/or experimental technology advance.

This distinction between actual experiments and "theoretical experiments" (a term with an interestingly semi-contradictory nature) comes up in other areas of physics as well. For example, Scott Aaronson's essay "Why Philosophers Should Care About Computational Complexity" puts a slightly different spin on some related ideas, and argues persuasively that philosophers should take seriously the practical limitations on what experiments and calculations could actually be performed in the real world.