The first issue is security. Loads of text editors create temporary files (.swp, .php~, etc) in the same directory as the file you're editing. These can render as text instead of going through php, thus revealing things like database secrets and such. Now, this problem won't always occur and can be configured around.
However, perhaps the bigger problem is that this is simply a bad workflow when a better one exists. For "live editing", run a local webserver. Don't expose your playing around to the world, potentially breaking your site for others. Heck, if it's a static site, just run "python -m SimpleHTTPServer" if you like. Second, for deploying, you should have a better process than "sync this directory". Specifically, you should have git hooks that handle deployment if you want don't-think-about-it instant deployment. Preferably you have CI that deploys after it passes a few tests. Using git also has other benefits like letting you have commit messages, easy rollbacks, different branches (which you can auto-deploy to test / dev / etc).
Basically, the problem with using Dropbox or BTSync to edit websites is that you rarely want to throw partial changes to the world, and file sync programs like to sync as often as possible. It's a cool idea, but those sync programs simply don't replace proper version control and CI.
I'd even prefer rsync if you're not using version control simply so you can control when it syncs and setup complex excludes / shell scripts to run around it. You could easily have a script that runs a minimizer and then rsyncs the output while having rsync exclude any potentially accidental files. You can't do that nearly as easily with the workflow the author is talking about.
I work with a largely non technical team on a site that is 95% built with a static site generator. They needed versioning but I couldn't convince them to take on the overhead of SVN or Git.
We sync the project using Dropbox, than deploy the build folder via FTP. We use folders for branches and Dropbox has limited versioning.
The biggest problem is that the people this is useful for probably aren't going to have a VPS or know how to install the btsync daemon. There are a couple of services that offer hosting for dropbox static sites like site44. I could see a Btsync equivalent being a popular service.t
There are problems with this method, but, for a hobbyist who hand codes a small static or very simple site, I think it would be fun. Magic internet stuff.
The interesting idea for me was not just auto-syncing files onto a remote server, but onto a web server. It's obvious once you think of it, but I hadn't thought of it before.
I think both use dropbox or btsync for this is a little weird. I wouldn't do it. It is not hard to publish sites.