Chosen Solution
When I replaced my screen I hooked up one of the cables in the wrong spot and it smelled like something was burning. Now the phone won’t turn on at all. What can I do?
The iPhone 6S is particularly sensitive to that mistake. In all likelihood, you blew the backlight filter and possibly even the entire backlight circuit. Either way, this will require a micro-soldering repair and it is not a DIY repair.