As Dan and Mark pointed out, the short answer to your question is NO. Quantum computing relies only on the mathematical framework of QM---i.e., the part that's common to all interpretations, to whatever extent they are interpretations rather than alternative physical theories. If a theory predicts that quantum computing can't work, the theory must either deviate somehow from the framework of QM, or else add some new physical principle to the framework with new observable consequences---neither of which a mere "interpretation" is supposed to do.
On the other hand, one can also ask the further question: do some interpretations provide more insight into how a QC works than others? David Deutsch, one of the inventors of quantum computing, was motivated by the goal of making vivid the Many-Worlds Interpretation (of which he's a strong believer), and has argued for decades that quantum computing makes any interpretation other than a Many-Worlds one look hopelessly contrived. However, others working in quantum computing vehemently disagree with that claim, and say that we can understand a QC just fine from (e.g.) a Copenhagen, quantum Bayesian, or "shut-up-and-calculate" perspective. Probably the majority of QC researchers don't care about the interpretation debate, or regard it mostly as a source of amusement. Their main goals are (a) to build devices that work, and (b) to understand what we could do with those devices.
However, here I'll add my personal opinion that some interpretations---such as deBroglie/Bohm and its cousins---look quite contrived if we try to use them to understand quantum computation. Yes, certainly deBroglie/Bohm predicts that QC can work, since all of its predictions are the same as standard QM's. However, in any interesting quantum algorithm (like Shor's algorithm), the computational "work" is clearly done by unitary transformations on an exponentially-large, highly-entangled n-particle wavefunction---a situation that leads to intuitions very different from those suggested by one or two particles moving around in a potential. If you worked out the trajectories for the Bohmian particles in Shor's algorithm, they'd look like a comically-irrelevant sideshow to the main event, adding no explanatory value and just "tagging along for the ride." (See this question for more.)
Finally, for some interpretations, like the "transactional interpretation," I don't think it's ever been satisfactorily explained how they can account for quantum computation. But if so, then that's simply another way of saying that it hasn't been satisfactorily explained how they reproduce QM itself. See here for more.