Will DNA use an end-users internet connection even after they have completed downloading?

Yes, the DNA peer network relies on sharing user-contributed bandwidth, and will upload fragments of DNA-accelerated content to other users who haven't yet downloaded those fragments. This trading of content pieces will occur while a user is downloading a piece of content accelerated by DNA, and for a short period afterwards. DNA only forwards pieces of the exact same file that was downloaded. It is never able to forward (nor is it aware or able to access) any of other files or private data on a user's desktop.