What can happen if I don't properly ground my equipment?
There are several "ground" issues for a station. Lightning ground from your antenna(e), and antenna ground plane are two on the outdoors. I assume your question is about the grounding within the shack.
Safety - a power ground will protect the operator from electrical (high voltage) malfunctions. Simply put, a ground to the "third pin" on the outlet or a convenient cold water pipe could save one from a nasty electrical shock if something malfunctions catastrophically.
RFI - this one is a little more complex. Depending on your feedline, antenna and outside grounding; you may get RF riding back into the shack. A good RF ground can prevent many of the RF interference and RF burns issues within the shack. What makes a good RF ground? You could ask ten operators and get a different answer from each. Some believe the electrical ground will suffice. Some believe a separate ground to an outside grounding rod is necessary. Yet others believe multiple ground paths are required. And there are some that do NOT use any ground as they believe it simply provides RF and lightning a path through the shack.
Regardless of the myriad of opinions, a "ground" of any kind within the shack is always a good safety measure to protect against power supply failures or other high voltage accidents, at the very least.