Why did my touch screen stop working after I replaced the screen?

Did you do the replacement yourself? If so, I would suggest attempting to clean the LCD connectors on the motherboard of the device. iPhone 6 Plus is infamous for "touch disease" after repairs, some of them even beforehand. If the touch does not restore itself, it may be the Touch IC chip on the device that has gone out on you. Some shops that offer micro-soldering may be able to do this repair for you.