Probably not a problem. You didn't mention which software you were concerned about, but some of the ones that are the most fussy about licensing include those from Adobe and Microsoft. Do you have any of those?
Adobe explicitly states that you should deauthorize on the old computer before installing on a new one (or migrating to a new one). But any issues you might encounter really depend on which versions of the Adobe software you have. I have migrated over older versions of Photoshop and Photoshop Elements and they ran fine on the new computer, even with the old computer up and running at the same time. The newer ones might check to see if the same serial number is running on more than one computer and tell you to deauthorize one of them. But since you aren't actually running the software on the old computer now, this might be fine.
I recently migrated Adobe Photoshop CS4 from an older computer to a new one and it would not run on the new one until I completely uninstalled it there and then reinstalled from scratch. So you may need the install disks and serial numbers anyway.
With Microsoft Office (2011), the migration worked fine but it required re-entry of the serial numbers on the new computer. If you run MS-Office on both the old and new computers simultaneously, it may notice that and refuse to run both at the same time. Again, not a problem for you since your old computer is not operational.
In any instance, should you run into trouble, you can contact Microsoft or Adobe (or other companies for other software) and they will reset the licenses from their end so you can use the software on your new computer.
By the way, if you use iTunes, you can do something like that from the iTunes Store, where you deauthorize all computers in order to get the old one off the list of authorized computers.
