I was reading the lidocaine user's manual for dentistry use and according to it and I quote "Lidocaine in concomitant use with acetazolamide, thiazides and loop acting diuretics. The hypokalemia ("a low level of potassium (K+) in the blood serum") induced by these drugs, antagonizes the lidocaine effects" which seems a bit weird as a potassium deficit would hyper-polarize the membrane increasing membrane potential making it to hard to get to the threshold potential right?
The following is a suggested answer only -- and is NOT intended to be authoritative; since it is not clear from the literature on Lidocaine why its efficacy is reduced in hypokalemia.
While it is true that hypokalemia will tend to induce hyperpolarisation of cell membranes, which in the case of neurons inhibits the elaboration of the action potential (by increasing membrane potential), other effects should also be considered, in the first instance that the concomitant tendency for K+ to flow out of cells generally also implies the countervailing tendency for Na+ channels to promote the transport of Na+ into cells -- more channels operating with greater facility.
Since Lidocaine is a sodium channel blocker -- which prolongs the inactivation of voltage-gated Na+ channels -- then prima facie, a generalised condition of hypokalemia suggests that the associated operation of more sodium channels will tend to antagonise its action.
At the same time, since the Na+/K+ ATPase would be operating to maintain homeostasis of intracellular cations (K+ in particular) in the face of any such efflux of K+ from cells, this effect would likely be accentuated by virtue of the corresponding efflux of Na+ from the same cells; notwithstanding the findings cited in the study below  concentrating on myocardial activity, that ''hypokalemia significantly inhibits Na+/K+-ATPase pump activity causing intracellular Na+ and Ca2+ accumulation''. In that case, it is presumably the implication of the Na+/Ca2+ pump which complicates any effect on Na+ channels.
Additionally, the finding that hypokalemia inhibits the Na+/K+ pump would imply a tendency to recalcitrance in the opening and function of sodium channels which, since that would favour the action of Lidocaine (as a sodium channel blocker), contradicts the premise of the question.