I'll split this answer into two parts:

1. Email Validation in terms of double opt-in

2. Email Validation in terms of actual real-time verification using a web service / API

3. In my opinion, the practice of double opt-in is advisable when

• email is one of the most important means of communication between your business and its customers
• sensitive data (e.g. invoices) are being sent out via email
• crucial service notifications/announcements are being sent out via email
• if your product/website generally tends to attract spammers/abusers

If, however, your website does not make any special use of customer email addresses and relies on a system of simplicity, I'd skip the double opt-in, or at least rethink it, since it is one more step the user needs to take before being able to use your service (a matter of UX)

2. As to the use of real-time email validation on the back-end, in my opinion, every business should verify email addresses in real-time via an API.

For the simple reason that is mostly free (Free, Powerful Email Validation API), it verifies the actual existence of an email address, checks for misspellings and typos, checks for free & disposable email providers, and it does not require the end-user to do anything.

Any good bulk email validator will perform a number of checks to determine the probability that an email is valid or not without actually sending an email to the email address it is checking. While no method is foolproof, doing bulk email verification is highly recommended for any serious email marketer to keep email deliverability high in order to maximize email ROI.

A good email verification tool can help determine which emails are valid, in bulk and at scale. A good email verification service is fast, accurate, and easy to use. If you want to send emails like the big guys, without the resources to remember to do bulk email verification or developers to implement complicated APIs, then I recommend a solution I founded called clearout. It's a true set-it and forget-it solution that takes the pain out of bulk email verification.