Does my participation in installing a new smart TV screen at home, which will be used for various purposes, both religious and non-religious, count as aiding in sin, and will I incur a sin every time forbidden material is watched on it?

As long as screens will be used to watch what is impermissible, then it is not appropriate to assist in their installation. The intention of watching good content does not permit their installation, because averting harm takes precedence over acquiring benefits, especially if the harm is predominant.

It is incumbent upon whoever installed them to strive in advising their owners and warning them against watching what Allah has forbidden, alongside repenting to Allah for assisting them, with remorse and a firm resolve not to return to such an act.