Many heroes other than Captain America have used his shield over the years, but perhaps no one wielded it better than the Hulk. This happened in 2014's Hulk series in the arc titled Hulk DOA. It was written by Mark Waid and illustrated by Mark Bagley.
The story starts off with Bruce Banner getting shot in the back of the head by a mysterious organization. Having knocked Banner out, the group performs brain surgery on him, but the anesthesiologist feels bad and wakes Banner up. Bruce turns into the Hulk mid-surgery and escapes. Unfortunately, he's also suffering brain damage from the incident and has no idea who he is. In order to protect him, SHIELD moves Banner to a small town with SHIELD Agents, but his relative peace is interrupted when the bad guys finds him there.
To attack the town, the group created a mindless version of the Abomination programmed like a drone to endlessly go after the Hulk. With the Abomination and Hulk engaged in a brutal fight that tears the town apart, the Avengers show up and try to help. During the scuffle, Captain America loses his shield. As it turned out, Hulk had stolen it and was using it as a weapon to punch Abomination. Also, in order to defeat him, Tony Stark rigged a teleportation device to the shield, which was set to activate upon impact. Then Hulk threw the shield, as Cap often does, and teleported the Abomination out into space.
The Hulk is often known for his ruthless hand-to-hand fighting, so it was interesting to see him fight with such a tactical weapon. He wielded the shield essentially like a pair of brass knuckles. It also looked very small in his hands because fans are so used to seeing the shield being worn by a person of regular size rather than a giant Hulk. What's even stranger is that he was able to fling the shield like Captain America, which takes a certain degree of skill rather than just brute strength.
It's also important to note that this story took place while Bruce banner was suffering from brain damage. Despite Banner's intelligence, the Hulk isn't known for being the smartest hero. His whole appeal is about going on a mindless rampage with his fists. So it's bizarre that he actually wielded a difficult and obscure weapon at a time when his mind wasn't working at full capacity. It's almost as if the Hulk has fighting built into his DNA, so that he can instinctually use whatever's around him in an expert manner. In this case, that happens to be Captain America's shield.