If we place the Copenhagen interpretation within the broader class of objective collapse interpretations then yes, it is possible to 'falsify' the Copenhagen interpretation.
These interpretations claim that at all times there is a chance for a superposition state to collapse into (or towards) one state of the superposition or the other. The probability of this collapse is related to the "bigness"* of the superposition. Thus, if you create a superposition of a large rock being in two places at once it will collapse very quickly, whereas a superposition of an electron in two places at once will hardly ever collapse. Of course, once the electron is measured it is now entangled with the measurement device which is "big". This means as soon as the measurement is performed it becomes highly likely that the system will collapse immediately.
This postulate of "collapse" requires an extension to the unitary time evolution of "orthodox"* quantum mechanics which Many-worlds interpretations take very seriously. This extension means that the laws of physics for the objective collapse models are different than the laws of physics of "orthodox" quantum mechanics. The difference should be physically measurable. In fact it is.
Conventional quantum mechanics says that if you create a superposition state and this superposition doesn't interact with anything then it will persist forever. The objective collapse models say that even if the system doesn't interact with anything the system will still collapse, probabilistically, after some amount of time. This is a testable difference.
I propose the following double slit experiment. Put a particle in a superposition. Pass it through a double slit. Let the superposition propagate trough space for time $T$. Measure the location of the particle. Repeat many times. If the superposition persists then looking at all of the results you will see an interference pattern. If it does not then you will see the interference pattern wash away and the contrast decreases.
Now, vary the time $T$ and repeat.
Now, vary the mass $M$ or size of the particle and superposition state
Orthodox quantum mechanics says that as $T$ or $M$ is increased we should see no change in contrast. The objective collapse models say that the interference contrast will decrease as $T$ and $M$ are increased. Thus these two particular interpretations can be experimentally distinguished.
Now, there is a VERY important point here. Above I have stipulated that it is important that the superposition does not interact with anything else (air particles, photons, magnetic fields, etc.) while it is in the superposition. This is because, due to decoherence, if the particle interacts with anything that will ALSO be a source of contrast reduction. This is why it is hard to perform these experiments in practice. It is very hard to make it so that the quantum systems don't interact with anything. And the bigger the system is the harder it is to ensure there is no decoherence.
So to perform the experiment above we must create the superposition and see if it decays as a function of time but we must also have controlled the experiment well enough so that we can rule out any normal forms of decoherence as causing the reduction in contrast. For example, There are researchers now creating bigger and bigger superposition. As they make the superposition bigger or wait longer they DO see a reduction in contrast. However, they are able to explain this as being due to typical experimental imperfections/decoherence. Thus, they have yet compared the two interpretations in a significant way.
However, that last statement is not quite right. By performing that experiment they have said something about the rate of this "spontaneous collapse". They can say that they have experimentally determined that whatever the rate of spontaneous collapse is (call it $\Gamma_{SC}$), they know that it is SLOWER than the rate of decoherence, $\Gamma_{D}$, that they can explain in their experiment.
They have experimentally verified the following constraint for their choice of $T,M,N$ etc.
$$
\Gamma_{SC}(T,M,N) < \Gamma_D
$$
"Orthodox" quantum mechanics would say $\Gamma_{SC}=0$ whereas objective collapse models say $\Gamma_{SC}>0$. The goal of these superposition experiments is to increase $T$ and $M$ (thus increasing $\Gamma_{SC}$) while decreasing $\Gamma_D$ to see if they can perform a measurement that breaks this inequality which would require us to definitely add the objective collapse laws to the laws of quantum mechanics.
As I've mentioned it is very difficult to control and reduce the decoherence rate $\Gamma_D$. We are still many orders of magnitude away from performing definitive measurements ruling out objective collapse models for macroscopic objects. It may be decades or more before we are able to control quantum systems well enough to try to make really really big (like gram scale) superpositions but people will continue to work towards it.
See This article for some information about research going on now into making bigger and bigger superpositions.
*Here bigness can mean the mass of the particles involved in the superposition, the number of particles, the distance of the superposition etc.
**Here I use "orthodox" to mean unitary evolution of the wave function only. No collapse. This may be in direct conflict with usage where "orthodox" quantum mechanics means the Copenhagen interpretation. That is why I clarify here. Perhaps it would be better to say "Everettian quantum mechanics".