My understanding is that the Mongol rulers never learned Chinese, and spoke almost exclusively Mongolian at court, and in their daily lives. Was the aristocracy of China similarly forced to learn Mongolian? How much did it percolate down into the vernacular? Were official government documents written in Mongolian?
/u/Jasfss actually touched on this in a fairly recent AskHistorians podcast on the Yuan Dynasty. The relevant portion starts about 41 minutes in and starts by talking about the ethnic-based laws, which were key to the kind of segregation that occurred during this period.
Just to sum up, the Mongol ruling class did not adopt Chinese language or writing, instead relying on intermediaries. In particular, bilingual Uighurs acted as the go-betweens for the Mongols and the Han, though not always as the most honest brokers.