Characters Who Weren't Kings/Hands Who Sat the Iron Throne

Behind her loomed the Iron Throne, its barbs and blades throwing twisted shadows across the floor. Only the king or his Hand could sit upon the throne itself. -AFFC, Cersei V

Rhaenys Targaryen & Visenya Targaryen

Aegon's sisterwives (or wisters). I think this is pretty cool. Outside of the whole sibling marriage thing, he saw himself as basically first among equals:

The "rule of six," now part of the common law, was established by Rhaenys as she sat the Iron Throne while the king was upon one of his progresses. -TWOIAF, The Targaryen Kings: Aegon I

and:

It was the king’s custom to bring one of his queens with him wherever he traveled, whilst the other remained at Dragonstone or King’s Landing, oft as not seated on the Iron Throne, ruling on whatever matters came before her. -Fire & Blood I, Three Heads Had the Dragon: Governance Under Aegon I

Rhaenyra Targaryen

This makes sense it would happen (during the First Dance of the Dragons) as Rhaenyra had finally secured what she wanted:

In the fullness of her victory, Rhaenyra Targaryen did not suspect how few days remained to her. Yet every time she sat the Iron Throne, its cruel blades drew fresh blood from her hands and arms and legs, a sign that all could read. Septon Eustace claims the queen’s fall began at an inn called the Hogs Head in the town of Bitterbridge on the north bank of the Mander, near the foot of the old stone bridge that gave the town its name. -Fire & Blood I: The Dying of the Dragons: The Red Dragon and the Gold

Trystane Truefyre

This might be cheating as technically Trystane was crowned, but Trystane Truefyre sat the Iron Throne for a short period in the Dance:

Royal decrees also issued forth from atop Aegon’s High Hill, where Ser Perkin’s catspaw Trystane sat the Iron Throne, but those were of a very different nature. -Fire & Blood I: The Dying of the Dragons: The Short Sad Reign of Aegon II

Jaime Lannister

Likely abandoned foreshadowing of Jaime becoming king, we have Jaime sit himself on the Iron Throne after killing Aerys:

"I cannot answer for the gods, Your Grace … only for what I found when I rode into the throne room that day," Ned said. "Aerys was dead on the floor, drowned in his own blood. His dragon skulls stared down from the walls. Lannister's men were everywhere. Jaime wore the white cloak of the Kingsguard over his golden armor. I can see him still. Even his sword was gilded. He was seated on the Iron Throne, high above his knights, wearing a helm fashioned in the shape of a lion's head. How he glittered!"

If interested: Hear Me Roar: Jaime Lannister's Left Hand

Euron Greyjoy

Cheating again here, as this is not only a shade of the evening induced vision that hasn't happened, Euron considers himself King/God/etc.:

No godless man may sit the Seastone Chair!”

“Why would I want that hard black rock? Brother, look again and see where I am seated.”

Aeron Damphair looked. The mound of skulls was gone. Now it was metal underneath the Crow’s Eye: a great, tall, twisted seat of razor sharp iron, barbs and blades and broken swords, all dripping blood.

and:

He saw his brother on the Iron Throne again, but Euron was no longer human.

If interested: Future Characters on the Iron Throne

TLDR: A list of non kings/hands who have sat the Iron Throne. It seems that every character either was seen as equal with the king (Rhaenys/Visenya), considered herself to be the Queen (Rhaenyra), was actually crowned (Trystane), GRRM intended to crown him at some point (Jaime) or could crown him still (Euron).