The short answer is 'yes', it is professional. In fact, it's not just professional but desirable. And I base that answer not on opinion but on reviewing the research on the topic.
Let me explain.
I read this post a few weeks ago, and was intrigued not just by the number of answers but by the lack of consensus. Like others here, I'd assumed this question was now redundant, given how we've all long gotten over the whole 'look at me with a new iPhone' thing. But I'd begun wondering about it again after the default signature reappeared on my iPhone after a software update.
I'd hesitated to just change it. (OK, I procrastinated, as I was irritated that I had to do it again, having carefully set up my mobile signature a couple of years back.) Then I wondered if I was actually better off leaving it as it was. Which is why this thread caught my eye.
Rather than just add my two cents', I decided to see if there had been any research on the topic. It turns out that there has.
A few years ago, two social scientists at the University of Oklahoma tested emails with and without 'Sent from my iPhone' on over a hundred volunteers, to see how each affected perception of the sender. In particular, they wanted to test whether the line reduced the damaging effect of grammar errors and typos.
What they found was that, while such mistakes did indeed have a negative effect on perception, 'Sent from my iPhone' significantly reduced that effect.
The reason is that we're all programmed to look for cues that help us judge the people we're communicating with. In a nutshell, we want to know whether we can trust and believe them – a concept covered by an idea called Uncertainty Reduction Theory (first put forward in 1975). Mistakes in an email increase uncertainty about the sender, but other cues called 'high-warrant' cues can reduce it. 'Sent from my iPhone' is the second type of cue.
So those people here who've suggested it buys you leeway are spot-on: it repairs some of the damage that typos (or hurried, terse replies) do to your credibility.
The original Oklahoma study research paper is here.
I've also written up the research findings in more detail on my blog here.
I may have geeked out a little, but it strikes me as a fascinating area – not least because I've never come across these ideas in any business books (despite working in communication for about 25 years). If the people we email (etc.) are all driven by the desire to reduce uncertainty, then tweaking what we write to help them with that (by including these cues) can only improve our work relationships.
Adding something like 'Sent from my mobile – apologies for any typos or strange predictive text' seems like one easy, simple way to do just that.