The short answer is yes, larger trucks like moving trucks will be required to go through a weigh station in some states; AAA has a guide to weigh station requirements in Canada and the United States, and you should also ask the agency you are renting from.
You will not necessarily have to stop at the weigh station for an inspection, as modern weigh stations have rolling scales— or you may simply be waved through by the agent or officer. Sometimes, there is even a special lane for moving trucks. But it is better to be safe than sorry, as the ticket for failing to stop at a weigh station when required can be costly.
To generalize, most states require only 1) commercial vehicles, of which your rented moving truck or pickup would not be included, and 2) vehicles over a certain weight, again, which your rented moving truck or pickup is unlikely to exceed, to stop at weigh stations. Some explicitly include or exclude trailers, motor homes / recreational vehicles (RVs), or campers.
The most common limits are 8,000, 10,000 lbs and 26,000 lbs Gross Vehicle Weight Rating (GVWR). It is extremely unlikely that a motorhome or a moving truck of personal goods would exceed 26,000 lbs., which is a common level at which a commercial driver's license is required.
But many of the larger trucks you can rent from Penske, U-Haul, and so forth would fall under the regulation. The Movers.com Weigh Stations page offers the following table of approximate gross vehicle weights:
- 14-foot has GVW of 14,050 pounds
- 17-foot has GVW of 14,050 pounds
- 24-foot has GVW of 18,000 pounds
- 26-foot has GVW of 20,000 pounds