The Yuan case is interesting. The Mongol elites certainly spoke Mongolian, of course, and in fact the early Yuan emperors at least seem to have spoken little Chinese if at all. The Yuan created a system that separated the major ethnic groups, so that the Mongol elites would be educated in Mongolian, the Han Chinese in Chinese, and there were also schools where Turkic and Persian were taught. Initially, the Mongols pretty much refused to learn Chinese and the Han Chinese subjects were not allowed to learn Mongolian. In time, some Han Chinese would be admitted to schools where they could learn Mongolian. Mongolian was the official language of the empire, so the government bureaucracy would have employed many translators who could produce documents in Mongolian.

The Mongolian language had considerably less influence out west in the Ilkhanate and the Golden Horde, because even though the leaders were Mongols, many of the soldiers would have been Turkic speaking (so much so that contemporary Europeans called them Tatars, not Mongols), and the conquered populations would have been used to other languages (like Persian in the Ilkhanate). Witness the use of the name 'Chagatai' to denote a variety of the Turkic language spoken in the khanate. Mongolian would have been limited to chancery usage among the Mongol elite during the early stages, and would have soon been overwhelmed by Turkic (the native language of a large portion of the khans' armies and of many conquered peoples) and Persian (the regional prestige language).

Since the Mongols weren't big on imposing their language on their subjects, visible traces of Mongolian in the languages of those who were conquered are hard to come by. For example, East Slavic languages have plenty of Turkic or Persian loanwords dating from the time of the Mongol khanates, but I doubt you would find many direct borrowings from Mongolian. But in Korean, there are at least dozens of Mongolian loanwords, especially in hunting- or horse-related vocabulary. For example, young goshawks less than a year old are called 보라매 boramae, which is a combination of Middle Mongolian boro and 매 mae, the native word for "hawk" (the Korean dictionary I'm consulting doesn't give a gloss for boro).