DIAGNOSIS: This is almost always a software glitch, not a failing screen. A background process or memory leak is bogging down the interface.
FIX: Perform a soft reset. Press and hold the physical button on the front of the thermostat (below the screen) for about 10-15 seconds until it reboots. You'll see the ecobee logo. This clears the cache without erasing your settings or schedule.
IF THAT DIDN'T WORK: If the lag persists after a reboot, check for a firmware update. Go to Settings > About > Version. If an update is available, it will prompt you. Ensure the thermostat is connected to Wi-Fi to receive updates.
HEADS UP: A poorly fitted or after-market wall plate can cause static discharge that interferes with the capacitive touchscreen. Make sure the thermostat is snapped securely and fully onto its wall plate.