Why doesn't Wolverine die when he loses his ability to heal?

Most of Logan's death-defying experiences are excused by his regenerative ability or the added strength of his adamantium skeleton (added in 1979 per the movie canon timeline). When one is inactive, the other picks up the slack, so to speak.

In The Wolverine I believe the audience is meant to understand that his skeleton is deflecting and/or softening impacts. Other movie tropes are also at work, of course, such as the super strong protagonist who can survive lethal injury.

The bug itself is a trope, the symbiote, which even in destructive cases usually wants the host to live as long as possible for maximum benefit. I can only speculate as to whether the bug somehow kept Logan alive or allowed traces of his regeneration to become active when he would have otherwise died, or if keeping him alive until he was on the operation table was in Viper's interest, etc., but given Viper's revealed plans to extract and harness Logan's ability itself, it seems reasonable.

Also don't forget the adamantium in his body its not normal human bone it's the adamantium (the only thing that it can fight against it is Captain America's shield and adamantium itself). Hence he might have lost his healing ability but his bones are strong as hell and while fighting that big robot guy he was very careful (watch the scenes closely).