Is it reasonable to classify logical propositions that rely on deduction and are non-falsifiable as being inherently not worthy pursuing or does this just applies to inductive reasoning?
It depends. Popper's falsifiability clearly discusses a way to define the nature of a fact, rather than the reduction of any logical argument.
We can claim that logical arguments that are non-falsafiable are non-scientific from an objective perspective:
These factors [Marxism, astrology] combined to make Popper take falsifiability as his criterion for demarcating science from non-science: if a theory is incompatible with possible empirical observations it is scientific; conversely, a theory which is compatible with all such observations, either because, as in the case of Marxism, it has been modified solely to accommodate such observations, or because, as in the case of psychoanalytic theories, it is consistent with all possible observations, is unscientific. For Popper, however, to assert that a theory is unscientific, is not necessarily to hold that it is unenlightening, still less that it is meaningless, for it sometimes happens that a theory which is unscientific (because it is unfalsifiable) at a given time may become falsifiable, and thus scientific, with the development of technology, or with the further articulation and refinement of the theory. Further, even purely mythogenic explanations have performed a valuable function in the past in expediting our understanding of the nature of reality.
As Popper represents it, the central problem in the philosophy of science is that of demarcation, i.e., of distinguishing between science and what he terms ‘non-science’, under which heading he ranks, amongst others, logic, metaphysics, psychoanalysis, and Adler's individual psychology. Popper is unusual amongst contemporary philosophers in that he accepts the validity of the Humean critique of induction, and indeed, goes beyond it in arguing that induction is never actually used by the scientist. However, he does not concede that this entails the scepticism which is associated with Hume, and argues that the Baconian/Newtonian insistence on the primacy of ‘pure’ observation, as the initial step in the formation of theories, is completely misguided: all observation is selective and theory-laden—there are no pure or theory-free observations. In this way he destabilises the traditional view that science can be distinguished from non-science on the basis of its inductive methodology; in contradistinction to this, Popper holds that there is no unique methodology specific to science. Science, like virtually every other human, and indeed organic, activity, Popper believes, consists largely of problem-solving.
Popper, then, repudiates induction, and rejects the view that it is the characteristic method of scientific investigation and inference, and substitutes falsifiability in its place. It is easy, he argues, to obtain evidence in favour of virtually any theory, and he consequently holds that such ‘corroboration’, as he terms it, should count scientifically only if it is the positive result of a genuinely ‘risky’ prediction, which might conceivably have been false. For Popper, a theory is scientific only if it is refutable by a conceivable event. Every genuine test of a scientific theory, then, is logically an attempt to refute or to falsify it, and one genuine counter-instance falsifies the whole theory. In a critical sense, Popper's theory of demarcation is based upon his perception of the logical asymmetry which holds between verification and falsification: it is logically impossible to conclusively verify a universal proposition by reference to experience (as Hume saw clearly), but a single counter-instance conclusively falsifies the corresponding universal law. In a word, an exception, far from ‘proving’ a rule, conclusively refutes it.
Every genuine scientific theory then, in Popper's view, is prohibitive, in the sense that it forbids, by implication, particular events or occurrences. As such it can be tested and falsified, but never logically verified. Thus Popper stresses that it should not be inferred from the fact that a theory has withstood the most rigorous testing, for however long a period of time, that it has been verified; rather we should recognise that such a theory has received a high measure of corroboration. and may be provisionally retained as the best available theory until it is finally falsified (if indeed it is ever falsified), and/or is superseded by a better theory.
However, using this philosophy, we can only claim that certain logical arguments are non-scientific, rather than non-useful. Expanding the debate to cover other philosophers of science, the nature of falsifablilty and the objective reality behind it may be a fuction of one's paradigm (Kuhn) or Research Programme (Lakatos).
It is not appropriate to apply the requirements of science to any logical argument, as not all logical arguments presume predictive power of the "real" world as part of their givens. When considering constructed realities (like computer protocols) it is even worthwhile to explicitly forbid considerations of the real, as they do not apply within the subjective and constructed (but logical) arguments therein.
Deductive arguments aren't non-falsifiable because arguments aren't either true or false. Deductive arguments are either sound, valid but unsound, or invalid. Here's an example:
(1) All men are mortal. (2) Socrates is a man. (3) Therefore, Socrates is mortal.
It's only the conclusion, or one of the premises that could meaningfully be said to be falsifiable. And of course, you could falsify (3) in one of two ways, which the argument makes clear: either by showing that all men aren't mortal, or by showing that Socrates isn't a man.
I don't see any reason to think that Popper would have a problem with deducing (3) from (1) and (2).