I am on Dreamhost VPS with root access. It runs Apache, and is hosting a site "www.example.com". At the same time, I am developing a Node.js web site, and binding Node.js to port 3456 (for example). So the Node.js site is accessible by typing "www.example.com:3456".
These are two distinct websites. I don't ever want users of the "www.example.com" accessing my Node.js website (which will be migrated to Nodejitsu after development).
Will I run into any problems with this setup?
I do not believe this will be a problem, unless one of your visitors happens to end up at port 3456. To mitigate this, you should think about writing your own small piece of middleware to whitelist your IP (thus rejecting anyone else). You can see an example at: http://www.hacksparrow.com/how-to-write-midddleware-for-connect-express-js.html. I'm sure you wont have a problem modifying this to your needs.