Touch Disease is a common failure on the iPhone 6 Plus.
At first, the phone will typically develop intermittent touch control failure. For some phones, a gray/white bar starts appearing at the top of the screen. Twisting and applying pressure in certain spots sometimes allows touch control to work for a short period of time, but eventually, the touch interface ceases to function entirely. This is colloquially known as Touch Disease. You can look at my profile for a link to a blog article on Touch Disease or search the web for more info.
As Mikkel suggested, you need to isolate a variable and either identify a known-good screen or test your replacement screen in a working phone.
If you are confident that your replacement screen is fully functional and there is no damage to the Digitizer FPC connectors on the logic board, then your phone probably suffers from Touch Disease. This is not a DIY repair.
The current best practice on repairing Touch Disease on the iPhone 6 Plus is to solidify the M1 pad (which is the LCM_TO_AP_HIFA_BSYNC line). That pad tends to lift, partially or totally when removing the Touch IC (Meson).
Jason from STS did a great video on this earlier this year (with a few follow up videos as well). I encourage you to watch them.