Dirac does not intend classical to mean non-quantum mechanics; he intends classical to mean pre-quantum mechanics1. So no, this says nothing about de Broglie–Bohm theory.
Dirac opens his paragraph with
The necessity for a departure from classical mechanics is clearly shown by experimental results.
Dirac is not talking about any theory that one might call "classical" today. He is talking about the theory of classical mechanics that existed before quantum mechanics was formulated. If you read his paragraph, all his criticisms are directed against that particular theory. For example, he says
the forces known in classical electrodynamics
if an atomic system has its equilibrium disturbed in any way and is then left alone, it will be set in oscillation and the oscillations will get impressed on the surrounding electromagnetic field.
These properties are specific to pre-quantum classical mechanics. His argument thus says nothing about de Broglie–Bohm theory. In fact, if I suspect that if you had asked Dirac, he would have said that de Broglie–Bohm theory was not classical mechanics.
EDIT: I want to address Lubos's comments, because some people seem to be agreeing with him and downvoting me. Further, I was asked in the comments to explain why Dirac's criticism doesn't apply to Bohmian mechanics. If you think Lubos is a genius and take everything he says uncritically, go ahead and downvote. If you actually can reason critically and think indepedently, please read my rebuttal to Lubos's comments before downvoting.
Lubos is wrong about almost everything he says about Bohm's theory in his comments. Lubos claims that the pilot wave in Bohm's theory is observable. The pilot wave in Bohm's theory is exactly the same as the Schroedinger wave function, so if the wave function in quantum mechanics is unobservable, so is the pilot wave in Bohmian mechanics. Furthermore, Bohm's theory has exactly the same predictions as quantum mechanics, so it is clear that in Bohm's theory, a system will not "be set in oscillation and the oscillations will get impressed on the surrounding electromagnetic field," simply because it isn't in standard quantum mechanics.
The only way I can see to save Lubos' statements is to realize that Bohm's theory doesn't actually work relativistically2, and then to reason that because photons are intrinsicially relativistic, photons don't actually work in it. But even though Dirac was an incredibly smart man, it is completely ridiculous to claim that
- Dirac intended his argument to work for all non-quantum theories and
- he foresaw Bohm's theory and was clever enough to realize that his argument would be saved by the fact that Bohm's theory doesn't work relativistically.
Dirac was smart enough to know that you can't prove a theorem like this (i.e., no theory except quantum mechanics can explain observation) without stating the hypotheses. His hypotheses, which I am sure he thought he had clearly stated, were that physics behaves along the lines of pre-1900 physics ... e.g., the dictionary definition of classical as traditional in style and form. In 1930, I don't believe anybody was using "classical physics" to mean "non-quantum". (I'll retract this statement if you can find an example of "classical" clearly meaning "non-quantum" rather than "pre-quantum" before 1940.) They were using "classical" to mean "pre-1900 physics", i.e., physics before quantum mechanics and relativity came along.
1 A standard definition of classical is: traditional in style or form.
2 Bohm's theory's adherents may claim otherwise, but I don't believe anybody has demonstrated that one can have a Bohmian theory with Lorentz-invariant particle trajectories and general relativity, which means that relativistically, the theory has severe drawbacks.