Could the Heisenberg Uncertainty Principle turn out to be false?

In quantum mechanics, two observables that cannot be simultaneously determined are said to be non-commuting. This means that if you write down the commutation relation for them, it turns out to be non-zero. A commutation relation for any two operators $A$ and $B$ is just the following $$[A, B] = AB - BA$$ If they commute, it's equal to zero. For position and momentum, it is easy to calculate the commutation relation for the position and momentum operators. It turns out to be $$[\hat x ,\hat p] = \hat x \hat p - \hat p \hat x = i \hbar$$ As mentioned, it will always be some non-zero number for non-commuting observables. So, what does that mean physically? It means that no state can exist that has both a perfectly defined momentum and a perfectly defined position (since $ |\psi \rangle$ would be both a right eigenstate of momentum and of position, so the commutator would become zero. And we see that it isn't.).

So, if the uncertainty principle was false, so would the commutation relations. And therefore the rest of quantum mechanics. Considering the mountains of evidence for quantum mechanics, this isn't a possibility.

Addition

I think I should clarify the difference between the HUP and the classical observer effect. In classical physics, you also can't determine the position and momentum of a particle. Firstly, knowing the position to perfect accuracy would require you to use a light of infinite frequency (I said wavelength in my comment, that's a mistake), which is impossible. See Heisenberg's microscope. Also, determining the position of a particle to better accuracy requires you use higher frequencies, which means higher energy photons. These will disturb the velocity of the particle. So, knowing the position better means knowing the momentum less.

The uncertainty principle is different than this. Not only does it say you can't determine both, but that the particle literally doesn't have a well defined momentum to be measured if you know the position to a high accuracy. This is a part of the more general fact in quantum mechanics that it is meaningless to speak of the physical properties of a particle before you take measurements on them. So, the EPR paradox is as follows - if the particles don't have well-defined properties (such as spin in the case of EPR), then observing them will 'collapse' the wavefunction to a more precise value. Since the two particles are entangled, this would seem to transfer information FTL, violating special relativity. However, it certainly doesn't. Even if you now know the state of the other particle, you need to use slower than light transfer of information to do anything with it.

Also, Bell's theorem, and Aspect's tests based off of it, show that quantum mechanics is correct, not local realism.

In precise terms, the Heisenberg uncertainty relation states that the product of the expected uncertainties in position and in momentum of the same object is bounded away from zero.

Your entanglement example at the end of your edit does not fit this, as you measure only once, hence have no means to evaluate expectations. You may claim to know something but you have no way to check it. In other entanglement experiments, you can compare statistics on both sides, and see that they conform to the predictions of QM. In your case, there is nothing to compare, so the alleged knowledge is void.

The reason why the Heisenberg uncertainty relation is undoubted is that it is a simple algebraic consequence of the formalism of quantum mechanics and the fundamental relation $[x,p]=i\hbar$ that stood at the beginning of an immensely successful development. Its invalidity would therefore imply the invalidity of most of current physics.

Bell inequalities are also a simple algebraic consequence of the formalism of quantum mechanics but already in a more complex set-up. They were tested experimentally mainly because they shed light on the problem of hidden variables, not because they are believed to be violated.

The Heisenberg uncertainty relation is mainly checked for consistency using Gedanken experiments, which show that it is very difficult to come up with a feasible way of defeating it. In the past, there have been numerous Gedanken experiments along various lines, including intuitive and less intuitive settings, and none could even come close to establishing a potential violation of the HUP.

Edit: One reaches experimental limitations long before the HUP requires it. Nobody has found a Gedankenexperiment for how to do defeat the HUP, even in principle. We don't know of any mechanism to stop an electron, thereby bringing it to rest. It is not enough to pretend such a mechanism exists; one must show a way how to achieve it in principle. For example, electron traps only confine an electron to a small region a few atoms wide, where it will roam with a large and unpredictable momentum, due to the confinement.
Thus until QM is proven false, the HUP is considered true. Any invalidation of the foundations of QM (and this includes the HUP) would shake the world of physicists, and nobody expects it to happen.

The Heisenberg's relation is not tied to quantum mechanics. It is a relation between the width of a function and the width of its fourier transform. The only way to get rid of it is to say that x and p are not a pair of fourier transform: ie to get rid of QM.