Can Azure host ColdFusion?
Solution 1:
The newly announced Azure VM role does allow you deploy a Windows VM configured anyway you like and you can RDP to this, so in that respect yes you could do this.
However, Azure vm's do have some difference to ordinary Hyper-V or VMWare VM's. You can't just spawn a new VM in Azure, RDP into it and configure it the way you want. Instead with Azure you create your VM on you own Hyper-V server and configure it how you want, then you configure a Golden Image that you upload to Azure and get's deployed to your VM. At any point Azure can shutdown your VM and create a new one from your golden image. So you need to be in a position where you can create this golden VM how you want locally first.
There is an added complication in that unlike with the Web and Worker role, the VM role is not manager by MS but by you and so you will need to apply any updates patches etc yourself. Again this means you need to not only update your running VM, but your golden image.
Finally you cannot really store any data on your VM, as MS does not guarantee the safety of this data, if your VM gets recreated from you master image, all you get back is what is on the image. Hence you need to store all you data in something like an azure database.
Solution 2:
A bit of an updated answer: Azure Virtual Machines can host Cold Fusion. Unlike the (now defunct) VM Role, Virtual Machines are created in the cloud. As a baseline, you can choose from several Linux and Windows variants, and then build up from there.
Once your VM is set up the way you want, you can image it, making it available in your Virtual Machine gallery.
One nice thing about Azure Virtual Machines is the per-minute pricing model, where you can stop your VM's any time within the hour - it's not rounded up.
More info on Virtual Machines: here.
Adding a bit more to this: There's no need to shift from MySQL to SQL Database (unless you want to). MySQL runs without issue in Azure, whether directly in Virtual Machines or via a hosting partner (through the Marketplace in the portal).