If you ever receive this error when trying to launch QTP:

QTP license error "Unable to contact the license server"

Here is the fix:

Right mouse click on My Computer --> Properties --> Advanced --> Environment Variables --> User Variables
you should see LSHOST variable, check if the IP or machine name fits the real QTP license server.

Since this was only failing on my computer and not my coworkers, I wrote down what their LSTHOST variable was and added that to my computer. I was then able to launch QTP.

Hope this helps!