From my basic searching I wasn't able to find anything specific, and I don't believe I'm likely to find anything as the details aren't released to the public. There's some basic information out there from this article (https://www.nytimes.com/2009/01/23/us/politics/23berry.html) though that lets us draw our own conclusions:
"First, only a select circle of people will have his address"
There's a bit of security-through-obscurity happening here where if you wanted to send Obama a phishing email, you'd have a hard time figuring out even the easy part (where to send the email address). As a security engineer we avoid relying on security-through-obscurity measures like this, but it can absolutely be helpful as part of your defense in depth. Probably not practical for a company because this information is a little more public - using email@example.com allows everyone to know each other's email/memorize their own, and the names of the employees at your company is more or less public information and easily found.
"Second, anyone placed on the A-list to receive his e-mail address must first receive a briefing from the White House counsel’s office"
This is interesting, but non-technical and non-practical for a company to implement in the vast majority of cases. Employee training/awareness might be the closest parallel.
"Third, messages from the president will be designed so they cannot be forwarded."
There are a few methods that email providers implement that prevent email forwarding. This may not seem like a particularly strong mechanism because anybody who receives a message can manually copy/paste and forward it themselves, but it's important to keep in mind that a lot of security controls protect against ignorant and forgetful users rather than malicious ones. This is a simple defense-in-depth measure that reminds users not to forward sensitive mail.
"he had to agree to use a specially made device, which must be approved by national security officials."
From this I'm inferring that Obama doesn't have full control of his device aka isn't a root user. This is something you see companies doing every so often when they provision mobile devices for their employees - you want your specialists in IT to have full control over the device and not allow your users to disable a safety mechanism or have their password associated with root privileges. If you're provisioning a phone for an employee there should be no cases where they'll require root/privileged user access, so just access to the phone, messenger, and internet apps may be enough.
"“It’s a pretty small group of people,” Mr. Gibbs said, explaining who would be allowed to e-mail the president."
I'm guessing that not only does this small group get access to Obama's email address, but a mechanism in the phone will drop any connections/reject any messages from addresses not in an approved whitelist. Would definitely be useful implementing this in a company if you have a business context where you only expect your users to receive messages from a known list of people.