If you're engaged in billable time practices, responding to e-mails is little different than phone calls, faxes or letters. It's all about the time you spend engaged on the work for the client.
Most of the emails I get are from people who want my money (vendors/businesses). But I get the occasional email from someone who wants to give me their money (a potential client). I wouldn't want to do anything that would stop a potential client from sending me an email.
Two other concerns:
1. The practicality aspect. If I'm requiring bitcoin, then the sender needs to be bitcoin-literate, which not everybody is. If I'm requiring a credit card, then the charge is too small to justify the merchant charges.
2. The branding aspect. Do I really want to be known as someone who charges for emails? I'm not sure that message would align with very many people's personal brand/image. I'd be more likely to wait on the sidelines until charging-for-emails is commonplace, then I'd feel comfortable joining in. But until that time comes, you might have a first-mover-disadvantage. This creates a problematic catch-22 for any SaaS/PaaS offering this kind of thing. (Few people wanting to be an early-adopter).
The answer is most likely no.
I think that goes for most people.
It makes a little more sense to charge for a guaranteed response as LinkedIn does indirectly...
And most people have potential clients, casual acquaintances etc that would enrich their life by contacting them, but certainly wouldn't pay 0.1 BTC to do so, so I'm not sure I'd ever stick pay-to-email as a contact on a business card or website. Those people whose email addresses people would happily pay for tend to be able to pay for assistants to screen their primary contact email anyway...
Charge who? Your clients? Businesses trying to contact you? Everyone?
How? Do they have to pay before they can even send it? You send them a bill afterwards?
This is how a solution could look like: http://wrte.io