Desktop PC Sounds High/Low Beeping and Turns Off During Boot

Problem:
After turning on desktop PC, it makes a high/low beeping sound and then a few seconds later during boot up turns itself off.

Solution:
In this specific case, the warning was an overheat warning. The CPU was overheating and shutting itself down for safety. The CPU heat sink was found to be slightly loose. Removing the CPU heat sink and removing/reapplying thermal grease resolve the issue.