The Teamviewer version was not specified.
Older versions allowed clipboard sharing (including files) by default. Worse, the clipboard sharing did not have any indication of being used, so one can copy files to a remote computer (possibly on Startup locations) without anyone noticing.
There's a risk that a program may have been copied over to the machine being remote controlled. This doesn't have any immediate effects, but any malicious payload will get activated on next boot. One can also replace files that are periodically used by services. So yes, the machine may be infected after a normal reboot.
Running a live CD and doing a manual check may be the best way to go. A virus scan may miss obfuscated files, or the malicious payload simply isn't recognized by the scanner. Realistically, there's a lot of attack options once one has write access to a machine (e.g. replacing commonly-loaded driver files, replacing files used by common services), so a manual check might not even be feasible.
Using the approach above, the router may be infected in theory, though I highly doubt that unless you're up against a persistent, dedicated threat.