Infectious disease expert Jeffrey Starke, a paediatrics professor at Baylor College of Medicine, tells The Wall Street Journal that even though there might technically be germs in an unwashed coffee cup, it's fine because they are yours. These germs, even if you're sick, won't hurt you again if you drink out of the same mug. "Very few viruses can exist for more than an hour on an inert object" Starke said.
The only time I would take that advice is if the only option available to you was to use a shared dirty sponge from the office kitchen and the water was luke warm - in that case, doing nothing would have to be better.
Pro Tip: make use of the microwave to boil the water and rinse your cup. Maybe boil that sponge too!