Good question! In my LTL days refused loads were quite common. As an expedite driver I have had a few consignees refuse to unload my van, even though they had a forklift and ramp. In one case the customer got involved and jacked up the consignee (USPS) and eventually I got unloaded. Another time I drove about 75 miles from the consignee to a warehouse to unload.
I did deliver a load from St. Cloud, MN to Belvidere, IL. that did get refused, but not until after I delivered it and got about 20 miles away. The freight was a long, heavy crate, that contained a large shaft for a food blending machine. I delivered early evening. Two maintenance guys were waiting for me, eager to unload and install the shaft before morning. The BOL was signed and I headed up to the Pilot in Beloit for the night.
I was at the Pilot maybe 10 minutes when I got a call from dispatch to return to the consignee and pick up the freight. I had until noon the following day to bring the freight back to the shipper. It turned out the shipper had sent a shaft for the right side of the blender instead of the left side. The maintenance guys discovered the mistake after uncrating the shaft.
It was a nice 800 mile round trip and I ended up about 50 miles from home.