No. Trainz, when bought from Auran, is DRM-free. Now, TC3 may soon no longer be supported, so you might have to use whatever backups you have made, but games that are still officially supported should be available through your order history. TC3 is still officially supported, I think, up until September of this year.
Edit: While I don't know for certain if this applies to TC3, with TS2010 and TS12, you can also simply copy your existing installation to your new computer, so you don't lose any content or go through the headache of exporting and reimporting stuff. If you want to keep your file associations, you need to set them on the new computer to point to your installation (if you don't know how to do this, it might be easier to reinstall and then copy your old installation over the new) but the game will run just fine without being formally installed.
