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.
다이 and 쓰메끼리 are not standard words (which are in a dictionary) so we avoid using them. As you already mentioned, they are actually not Korean but Japanese vocabularies. If there were no Korean words for them then they would be able to become standard words (maybe), but there are already good Korean words for them so there is no reason for using those words. (받침 for 다이, 손톱깎이 for 쓰메끼리) Same goes to 벤또 which is a Japanese version of 도시락.
On the other hand, the word 담배 is not from Japanese but from Portuguese, Spanish and English (Tabaco and Tobacco). Not exactly like this, but something like Tabaco -> Tambaco -> Dambaco -> Damba -> Dambae(담배). It's true that Tabaco and Tobacco entered Japan and then came to Korea, but it does not mean 담배 is a Japanese vocabulary. Similar example is 빵(bread), which originated from a Portuguese vocabulary pão.
I'm not saying that using Portuguese is okay and Japanese isn't. The difference between 담배 and 쓰메끼리 is that 담배 has been well integrated into Korean language during several hundred years, but 쓰메끼리 has very short history. When there are newly introduced vocabularies from foreign languages, trying to keep original Korean vocabularies will make sense. It applies to not only vocabularies from Japanese but also from any other languages, like 스크린 도어(screen door) became 안전문.
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.
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.