Yes, the screen can be disabled indefinitely if you wish, whilst continuing to take measurements.
Simply disable 'power saving mode', which is the setting which instructs the device to light up in the morning (default 8am) then turn its screen off at night (default 9pm). With this mode disabled, the screen will not automatically power itself on or off, but will remain in either state indefinitely after you choose the setting (as long as it has access to a power source).
When the screen is off but the device is powered on, it will continue to take measurements.
Disable power saving mode:
Enter your device's settings menu via the center button > Performance > Power saving > Power saving mode > 'Off'