Which non-sinitic languages contributed to the wordstock of common chinese words before 1800? I am not interested in modern words like telephone, aeroplane, but words that were incorporated into the more frequent used part of the vocabulary from languages different than sinitic.
You have the entire stock of Sanskrit buddhist terms that have been incorporated to Chinese, like 菩萨, 佛, 涅盘 and similar terms. 苹果 is also Sankrit in origin, while 葡萄 and other fruits have been picked up along the Silk road.

Mongolian words such as 达赖 (as in 达赖喇嘛, where 喇嘛 is Tibetan), have also crept into the langauge through religion.

Cantonese and Vietnamese have had a lot of exchange throughout the years, but are mostly local.

Although Japan has greatly been influenced by Chinese from Tang and onwards, there has also been a large import from Japan to China. Words like 经济 and 社会 do not really make sense in Chinese (component-wise), but are Japanese in origin. A more modern example would be the redefinition of 萌 as a cute young person, like in 萌妹纸.

Refer to Marc's answer for Manchu and Russian influences; there are quite many loanwords from northern tribes, like 哥哥.