The pitfalls are likely to depend on your license type.
A single node locked license will not be good on a cloud VM as the license bindings are likely to get broken too easily.
I run a node-locked install of TC on a local VM. Which I have configured VERY carefully so that the ID's used for license binding shouldn't change. Even with that, I did manage to break the binding once. (Been using in a VM for a little over a year now I think?)
Other than that, it shouldn't really be any different to running on a local, networked, VM. Except that if using a floating license, you'll need to host it locally and configure it and the cloud VM's so they can see each other. Might be simple, might not. Depends if you have corporate firewalls and such in the way.
Have they finished transferring all the Azure VM stuff onto the new/updated portal yet? Last time I used it, the new portal was still missing a few abilities ....