Besides that, I think seeing other shirtless guys can also help you push harder. Seeing other guys working out shirtless and pushing themselves to build their bodies motivates me to push myself harder, to keep up with them. Even though we may not be lifting the same thing, I feel like I have to push myself at least as hard as the guys around me. Actually seeing the other guy's muscles flexing intensifies the effect. So, I recommend not just working out shirtless, but finding some other guys to workout shirtless together.
Some people will probably object that working out shirtless is not sanitary. That is nonsense. Your hands are much more likely to spread disease because you touch everything with them, including your own eyes and nose (mucous membranes are a common way disease organisms are spread). Yet, we would never think of requiring gloves to be worn! Instead, we suggest washing your hands frequently. Besides, with tank tops and shorts there is already plenty of skin contact with equipment, what makes the little extra that would have been covered by a tank top or cut up gym shirt less sanitary than the part which is already touching the gym equipment? The proper solution is to wipe down equipment between users, preferably with disinfectant. And I hope you shower after working out anyway!
If your gym doesn't allow guys to workout shirtless, ask them about why. Maybe they will reconsider. One possible solution where offending the opposite sex is a concern is to have designated male-only hours where shirts are optional. Many gyms already have a female-only time, in which case it is only fair. That said, I don't see any reason to require shirts when women are present. Women don't find shirtless men offensive at the beach or pool, and many do not find it offensive for men to lift shirtless in the gym, either. If your gym won't make shirtless accommodations, look into finding a gym that does. And if you actually run a gym, consider letting guys workout shirtless. It may even be a selling point to get members who prefer to workout shirtless. And remember, shirtless-friendly and family-friendly are not mutually exclusive.