After a lot of testing, and a lot of reading, I stumbled across this link to the iOS developer's website:
The key was this: "When specifying a mailto URL, you must provide the target email address."
Basically, safari for iPhone requires a target email address in the mailto. With most other browsers, you can leave it blank. To fix the issue, I inserted a dummy email address to make it work.
<a title="Send search link" href="mailto:remove_this@email.com?subject=Check%20out%20Bob&body=http%3A%2F%2Fwww.example.com%2FPeopleSearch%2F%3Fname%3DBob">Email your search results</a>
EDIT: This issue may no longer be the case for Safari browsers for iPhone version 7+.