I think he meant the HOA's insurance (including liability) for the common areas of the complex. If the HOA's insurance found out that an HOA board member or property employee knew about a residency violation, they might have grounds to cancel the policy, which would just end up costing the HOA money, which would raise the monthly dues of all owners.
Your (as the hypothetical owner of an apartment unit) insurance cares about whether you are occupying the unit as you said you would. I'm sure you can purchase a policy for primary residence, long-term leasing, and short-term leasing. If you're violating your policy, it could be declared void at any time, in particular if you ever make a claim.
If I were a good insurance agent, hypothetically, I also wouldn't write an insurance policy for short-term leasing in a property where I suspected such use wasn't allowed. It would probably be in my best interest to get proof from the potential insuree that such use was permitted.
As for that rebellious streak, we're talking about an investment here, and ways that people might be breaking voluntarily signed contracts in order to make a few extra dollars (the real-estate agent should offer the HOA terms to any potential seller, and then you sign the HOA contract when you purchase the apartment). Also, I prefer disruptions to business models, not disruptions to living arrangement. As mentioned elsewhere in this thread, there is no upside to the other residents, it's all a nuisance to them.