I hear you, OP, useless mentors like that are usually the first to publicize their "advisor" role in your company everywhere (resume, blog, etc).
If a mentor has spent considerable time with a company and agrees it's in good shape (traction, market prospects, etc), then why wouldn't they tap their contacts to help its founders fundraise/hire/fill whatever need? If you won't put your money where your mouth is, then don't call yourself a mentor/adviser.
Calling your contacts ≠ forcing them to do anything; it is up to them to follow up/not, invest/not. And every investor appreciates a good lead that's been vetted. To the 2nd comment saying they'd be scared to bug an Andreessen contact type, bullsh*t. As long as the startup is in the industry that he invests in and has great potential and metrics (per what he looks for in companies), why would he hate you contacting him with a potential moneymaker?? Either you're the intimidated, starstruck type (which is stupid, as he's a mere mortal like you) or you're admitting that you aren't a good assessor of company/team potential, in which case, you should NOT be a mentor.