The most important risk you face is if somehow mains voltage ends up contacting somewhere you get electrocuted and die.

There are 2 purposes of an earth ground: First it can be used as a reference for certain signals, such as microphones. Second, it can be used to protect against turning yourself into a sparker.

There is a clear separation between mains voltage and system voltages so it's typically not going to be a problem, but if a little wire ends up contacting the power supply case it can become energized and things start to get really bad.

Most of the electrical code where I live focuses on grounding as "Bonding", which is purely safety related for giving dangerous voltages a safe place to go.