If the 12V power supply powering the LED strips is the isolated/double insulated type, there is no need to connect the metal that the LED strip is attached to ground or anything from a safety standpoint (this may vary by country).
There may, however, be EMC considerations if for instance the power supply is unfiltered or you're using high frequency PWM to drive the LEDs, where connecting the common terminal of the power supply output to the metal backing could help reduce EMI.
Yes, there's no need to connect the metal heat sink to earth, if it's run off a double insulated power supply. If the power supply meets the relevant EMC standards, there should be no need to earth anything. It will have the required filtering an Y capacitor, to keep EMI within the legal limits.
Contrary to what many have stated in this thread, there's no requirement to earth all exposed metalwork. There are plenty of, perfectly safe, double insulated appliances, with unearthed, exposed metallic surfaces, on the market. They're designed so that all the live parts are double insulated from the metalwork. An example is a double insulated mains electric drill, with a metal chuck, which has some plastic gears/shafts in the the drive chain, to provide an extra layer of insulation from the motor's frame, which will be encased in plastic, even if there are other metallic parts on the case.