Question: My friend told me that 다이 (podium) and 쓰메끼리 (nail clippers) are not appropriate for use in a professional setting and that I should use the words 받침 and 손톱깎이 instead. What is unprofessional about these words? Is it because they are derived from Japanese? Other words like 담배 which come from Japanese are okay, I think.

Answer: 
다이 だい[台] and 쓰메끼리 つめきり[つめ切り·爪切(り)] were Japanese words that were borrowed when Japan ruled Korea from 1910 to 1945. Koreans made a lot of efforts to replace Japanese-origin words with Korean words. 

There are more words such as 벤또 (도시락, lunchbox), 가라 (가짜, fake), 뼁끼 (페인트, paint) as indicated in the [linked article](http://news.naver.com/main/read.nhn?mode=LSD&mid=sec&sid1=103&oid=002&aid=0000023898). 

As there are Korean words for all of them, it is not appropriate to use them in formal context unless you want to make a joke. A word like 빵 (パン, bread) and 담배 (タバコ pronounced as 타바꼬) came from Japanese, but they were originally Portuguese.
There are two main reasons.
First of all, 담배 or 빵 have been being used for more than a hundred year. Now it is considered a Korean word perfectly. In contrast, words like 쓰메끼리(손톱깎이), 다마네기(양파), 다이(받침) were imported when Korea was a colony of Japan. In that era, Japanese government probihited using Korean language to eliminate the nationality of Koreans. After independence, therefore Korean government tried to remove influence of Japanese language.
The other reason is, young people doesn't understand those words that much anymore. Teenagers may not understand what you say and ask you back, "What is 쓰메끼리?" Only old people use those words a lot.