As the title says, does the use of the Soul Forge in the Conflux web application cost something except time?
It takes time to upgrade a Dynasty Weapon, and it doesn't consume my Dynasty points. So I don't understand what's the catch?
All that it costs is time. The "catch", as you put it, is that they are getting you to log in to their website regularly, where they can serve you ads or other content that they want you to see.