Just kidding, you can throw it away. The payment system, which requires a credit or debit card, seems a weak link. Even if Burner Phone destroys their records, the banks won't. Also, you have to trust Burner Phone, and, presumably, whoever that dude is in their domain's DNS.
Update: Bitcoin support's coming soon, the creators say in a thread at Hacker News. [Thanks, Fredley!]