I was reading the Wolverine Origins comic where Captain America cuts Wolverine with the Muramasa blade. The cut, as it is revealed will not heal because of the blade. If Wolverine were to cut out everything around the wound, wouldn't his regenerative ability heal him? I'm currently unaware if this is done later in the comics as I haven't read that far.
No, Wolverine could not cut away the damaged tissue and regenerate normally. Once exposed to the weapon, any injuries he caused himself would only add to his misery.
The Muramasa Blade (we are discussing the second one, not the first) suppresses the healing factor of mutants such as Wolverine, Daken (Wolverine's son) and Sabertooth, so any injury caused by the blade could not be artificially healed.
Any injuries made with it took days to heal, indicating the healing factor of their mutant abilities were nearly completely suppressed. Creed had his arm cut off with the Muramasa Blade and was unable to reattach it (as he did when Wolverine cut off his arm with his claws).
The process by which it suppresses the mutant powers has not been clearly identified and has been speculated as a disruption of molecular structures preventing regeneration to "its magic". Given the blade's ability to cut almost any normal substance except memory-oriented constructs (psionic or magical) magic would be the most likely origin for the physical properties of the weapon.