If you mean influence from Mongolian, not much. Mongolian rule was pretty short, just under a century, and unlike with the Qing, there wasn't such an effort to make their language official. There were a handful of Mongolian words (but not only Mongolian) that showed up in dramatic performances at the time, but they were minimal and mostly just a gimmick.
One reason why this was happening though was that the Yuan court, along with banning the Civil Service Exams, enforced the use of the vernacular rather than Classical Chinese. However it was really their lack of interference and therefore lack of influence that characterises the linguistic scene at the time. They were much more interested in promoting the arts than directing the language. See Langlois, Rossabi, cited below.
There is of course also the 'Phags-pa script but this wasn't something the average person was using for Chinese, and it essentially died out completely with the fall of the Yuan. So this isn't really something that impacted Sinitic, though it has given us some hints for things like historical reconstructions. I wouldn't even be bringing up 'Phags-pa, but I'm certain if I don't, someone is likely to reply saying "hey you forgot about 'Phags-pa!" but actually, it isn't really relevant to your question.
However, if you want to talk more generally about influence, it was huge. The Yuan conquer of the Song is arguably one of the more important periods in the development of modern Sinitic varieties. Prior to this, the Song court was pushed by the Jīn 金 / Jurchens 女真 from Biànjīng 汴京 (today's Kāifēng 开封) south across the Yangtze River into Lín'ān 臨安 (today's Hángzhōu 杭州).
In addition to the court, the period saw larger waves of migration south. This had begun earlier when the Jurchens (a.k.a. Jīn 金) were fighting with the Liáo 遼 (Khitans) a century earlier. For people living in the Central Plains area, these were a solid couple of centuries where the North wasn't an ideal place to be, and many sought safety by heading south. This didn't always work for them, but what it did do is pushed these Early Mandarin dialects south into areas formerly speaking Wu, Min and a various other local varieties.
This along with the later Taiping Civil War was one of the major periods which shaped the current makeup of Sinitic languages spoken on the continent today, and is in part a major reason why Mandarin became so widely spoken in areas like Sichuan, Sunan and other parts of the near-South.
tl;dr: Mongolian as a language didn't directly influence Chinese, aside from a some short lived slang. The Mongols as historical event did, but as part of a series of wars in the North that led to mass migrations of Sinitic speakers south.
Sources:
Langlois, John D, Jr. (2014) China Under Mongol Rule
Rossabi, Morris (1988) Khubilai Khan: His Life and Times
edited to correct lazy mistakes