Teamviewer does NOT support taking screenshots or remote control without root. The only Android devices it does manage to pull this off is on Samsung devices
Update 2019 June: Now it supports a slew of different Android OEMs beyond Samsung. Notably, Google itself is absent (So no remote control on Pixel series, just screensharing)
This Teamviewer press release shows how it's done. It basically installs this Samsung specific addon from the Play Store. As a background, Samsung got this working to counter Amazon's Kindle Fire's new feature called Mayday. Technically, Samsung basically gave Teamviewer a bunch of special pseudo-root-level certificates and signatures so the app can work seamlessly for everyone (by having draw-over-other-apps and make-interactions-on-users-behalf access via the Accessibility APIs presumably).
This is how it looks on non-Samsung non-rooted devices [left] and how it asks for Super User access on non-Samsung rooted devices [right] (Click through for original resolution):