I believe that the imperial household retained use of the inner court for a couple of years and that the republican government took over the rest of the complex but it would be many years before an official body would curate the entire palace.
I can't easily find good accounts of the period between 1912 and 1933. Such a large structure in the middle of one of the largest cities in the world must have attracted quite a lot of attention and interest.
So:
What happened to the (presumably) massive staff?
Who maintained the palace complex?
How expensive was it to maintain?
Did anything valuable go 'missing'?
Were there any plans to take it over for other governmental functions?
Were there any plans - wholly or in parts - to sell it? Destroy it?
After the coup at 1924, the Last Emperor was evicted to Tianjin together with his staffs, Empress, and concubine.
During the Last Emperor stay, there had been reports of palace eunuchs surreptitiously selling smuggling several items to antique shops. When the Last Emperor ordered audit, though, fire started and consumed the Palace of Establishing Property where most of Emperor Qianlong's collection was stored. The Last Emperor argues that the fire was started by the eunuchs to hide their smuggling, but the truth is not known.
During Cultural Revolution, parts of the outer court was ravaged, including the two golden jars and tripods in front of Hall of Supreme Harmony. Nowadays you can only see traces of gold on the otherwise black surface. It was saved, though, by Zhou Enlai, who sent troops to guard the city (albeit rather late).
Parts of the collection was evacuated when Japanese invaded Beijing. Some of the valuables are still kept in Taiwan, moved under orders of Chiang Kai-sek.
During 1912 to 1924, the outer court was used for public use while the inner court was occupied by the Last Emperor. During Hoxer Rebellion, though, under treaty, it was occupied by foreign troops.
So far no voluntary plan either to sell it, though from 1949 to 1960s there were various proposals were made, including creation of public park or transport interchange or places of entertainment. The throne in Hall of Middle Harmony was dismantled during this period. Thankfully, the site was declared UNESCO Heritage Site, preventing further destruction.
Source: Forbidden Palace Tourist Center.
N.B. If the source must be books, sorry because I learnt this all from various Chinese tour guides during my stay in China. I used to guide foreign tourists to Forbidden Palace for a fee. Please inform me if this is not permissible and I'll gladly delete my post. I'm not too sure.