As an attorney, I'm not aware of any specific case where it's come up, but it almost certainly has at some point. I don't practice in personal injury or car accident cases, but I used to do similar and I know many attorney who do. Here's how it probably would play out:
Overloaded vehicle causes an accident. Someone in other vehicle is severely injured or killed. Injured person or estate of decedent sue.
Attorneys representing the plaintiff work hard to prove their case. If there are photos of the accident scene (there probably are), they may catch an inkling that the overloaded vehicle looks like it was hauling a heavy load and use that to prove more negligence on the part of the owner/driver. They may not have concrete proof of a scale ticket, but they will argue the evidence indicates the vehicle was overloaded.
But how much that all comes into play depends on whether the overloaded vehicle was just some regular individual or a commercial/company car.
If it's just a regular individual who's driving the overloaded vehicle, it ultimately may not matter. Very seldom (unless that individual is really rich and acting really negligently) would the plaintiff's attorney go after anything beyond the limits of the insurance policy. Going after more than that gets to be a PITA to collect anything. That said, if the insurance company decides the owner/driver isn't guilty of negligence so much as they are of intentionally doing something wrong, it may deny coverage. In that case, the owner could be screwed.
If the driver/owner of the overloaded vehicle is doing so on behalf of a company, however, that all changes. If it's a "company" car for an aftermarket parts manufacturer, or if its a commercial semi, I can guarantee you the vehicle's load will certainly play a part if the plaintiff's attorney gets an inkling that was the case. Those cases can be and usually are big money cases, because you are not limited by the few hundred thousand dollars of a standard insurance policy but instead are suing an entire company, which typically has millions in insurance coverage.