I haven't actually done the transition (always been on colocated platforms and just recently a small shared account) but can give you a heads up on what to expect.
It really depends on what kind of dedicated server you have - does the hosting company provide you with a server that has a fresh install of the OS and leave the rest of the config to you, or do they provide a ready-to-go platform that only needs your site content to get up and running? My response is from the start from scratch approach (more like colocation I guess) but still may provide some ideas for you.
Unless you are super-prepared and have covered _every_ base, there will be some bumps with the transition. The key is to test, test, and test until the cows come home.
The thing to remember ***ut shared platforms, in most cases, is the hosting company manages the infrastructure - OS, Webserver config, databases, developement platforms, security, backups, just to name a few. With a dedicated setup, you would might now be responsible for some of that, save for backups maybe. For sites on a shared platform, all you generally manage is your site - and not much else, as you rely on the hosting company to deal with the nitty-gritty details like ensuring php support is up and ready, virtual host configs and that backups are taking place.
My recommendation would be to try and duplicate, to the best of your ability, your current shared hosting environment on a server in your home or office. If that means starting from scratch, so be it, as it will aid in your management experience of your future dedicated platform, unless you feeling like paying the hosting company for making a Apache change or something trivial like that. As well, if this means setup of database and mail server, well, good luck. If you don't manage those services yourself currently and plan too in the future it's something to take into account.
If you don't have experience managing a dedicated server, or no one on your team does, it will be a challenge. You will run into issues that you can't solve and may have to ask for outside help - but that's fine because not everyone knows everything
If you are on the open source side of the world it's alot easier as the support community is extensive and you can get answers or ideas fairly quickly, with accuracy too.
Hope this helps. As Chad said, I'm sure more people will offer feedback from firsthand experience. If you have any specifics ***ut your setup I wouldn't mind hearing ***ut them - just to see exactly where you are coming from.
Disclaimer: I don't actually work at a hosting company, so... take it as such.
regards,