The chair was used by Kang to explore Captain America's timeline during World War II as he witnessed Cap's battle against HYDRA and Red Skull and Steve's fall into the ice after Bucky pushed him to save his life and then perish in the explosion.
However, when Kang watched the battle, Bucky was getting suspicious about something or someone is watching them.
Kang used his chair to attack the Avengers and transported them to future Earth's ruins where he told them that Captain America's responsible for all of the destruction. He demanded them to surrender the World War II legend to him but refused and fought him.
As Kang fought the other Avengers, Iron Man had gained access of his chair and explained to Kang that a portion of the time chair's physical system was based on Stark Industries technology. Stark then transported them back to their own timeline. Kang tried to attack Iron Man, but was stopped by the Time Chair's force field.
After his defeat, Kang ordered Damocles to recall both him and the time chair, sending them both back to the flagship. He then closed off the Star Industries backdoor exploited by Iron Man, and launched a full invasion on the 21st century Earth. The invasion was stopped by the Avengers, however, and Damocles was captured by S.H.I.E.L.D. and eventually became the headquarters of S.W.O.R.D.
The fate of the time chair following Kang's failed invasion is unknown, though it was likely under S.W.O.R.D. custody before it was destroyed along with Damocles itself when the ship was compromised by Skrull infiltrators.