It should be noted that some scholars are not in agreement on the origins of certain calques or loanwords borrowed from the Chinese. One argument is that since there are so many different Chinese dialects with such differing tones, it's almost impossible to pin down whether one of them is the root.
Also, since so many Chinese expression have been first borrowed into another language, before being borrowed again into English, it's hard to determine where the calque has it's roots. Despite this, here are some examples of Chinese calques in English:
1. Lose Face and Save Face: These calques are rooted in the Chinese expression 丢脸 (diū liǎn): 丢 (lose) 脸 (face), and 留面子 (liú miànzi): 留 (save) 面子 (reputation). Anyone whose visited China or knows the Chinese culture can tell you that keeping up a good reputation and preventing shame from befalling your family are huge in China. The Online Etymology Dictionary says that the term "lose face" was first seen in English around 1876, while the term "save face" was seen in 1898, first used among British expatriates in China. Some dispute this expression as a Chinese calque.
2. Mandate of Heaven and Son of Heaven: Since ancient Chinese times, the emperor was also referred to as 天子 (tiānzǐ): 天 (heaven's) 子 (son). This allowed the emperor to rule with great authority with a "mandate from heaven" or 天命 (Tiānmìng): 天 (heaven's) 命 (fate). But it also allowed others to revolt and usurp the throne, by claiming that heaven had willed the revolt, because it wanted to shift the mandate of heaven.
3. Long time no see: This calque has its roots in the Chinese expression 好久不见 (hǎo jiǔ bu jiàn), which literally translated is 好久 (long time) 不 (no) 见 (see). However one source claims that this calque has roots in American Indian speech.
4. Brainwashing: This calque comes from the Chinese expression 洗脑 (xǐ nǎo) which literally means to wash (洗) the brain (脑). Many sources date this expression in English as appearing around the 1950s, during the Korean War. One source says it was first used in English to describe how the Chinese communists attempted to deal with foreign prisoners.
5. Paper Tiger: This calque comes from the Chinese expression for someone who has superficial bravado, but is actually quite weak -- like the image of a scary tiger that is actually made of paper. In Chinese, the expression is: 纸老虎 (zhǐ lǎohǔ), 纸 (paper) 老虎 (tiger). The expression has been used as early as ancient China, but appeared in English in an 1836 book about The Chinese by John Francis Davis, who defined the term's usage for "a blustery harmless fellow." Chinese leader Mao Zedong used the term to describe the United States in 1956. Some dispute this expression as a Chinese calque.
6. Running Dog: This term comes from the Chinese expression 走狗 (zǒu gǒu), which technically, would be: 走 (walking) 狗 (dog). The Online Etymology Dictionary says that this term was first recorded in 1937, from Chinese Communist Party phrases that describe people or organizations or countries that are the lackeys of imperialist and colonialist countries. The idea is that a "running dog" is one that runs at the command of its master. I guess "walking dog" isn't as poetic.
7. Snakehead: This term comes from the Chinese word 蛇头 (shé tóu), literally 蛇 (snake) 头 (head). It describes a criminal that for thousands of dollars, helps to smuggle people illegally into or out of a country.
8. No Can Do: The root Chinese expression is 不能做 (bù néng zuò), 不 (no) 能 (can) 做 (do). Admittedly, I couldn't find a strong source for this, but I think it sounds like a calque.
9. Grassroots: This calque either come from the expression 草根 (cǎo gēn) 草 (grass) 根 (roots) or 基层 (jī céng) 基 (base) 层 (layer). The expression is used to describe
10. Barefoot doctor: This calque comes from the expression to describe the thousands of workers trained in basic medical care that traveled to the rural countryside of China to treat farmers and peasants. The term in Chinese is 赤脚医生 (chìjiǎo yīshēng): 赤 (bare) 脚 (foot) 医生 (doctor). Some dispute this expression as a Chinese calque.