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Changing my screen on iphone 6s and left battery plugged in. Slight burning smell and now phone wont switch back on. Any ideas what might have blown?

On the 6s if you don’t disconnect the battery before connecting the screen your backlight circuit will be electrically damaged. The backlight circuit consists of the connector, filters, capacitors, diodes, coil and the driver IC. One or more of these has failed. No matter which one it will require a reputable repairer to fix the problem as it will require microsoldering.