를 is an object marker, but in spoken Korean, it tends to be dropped often. I'd be inclined to say 공부 열심히 하세요 because, without '를', it's obvious enough what the speaker wants the listener to do.

"공부 열심히 하세요" means "Study Hard". but, "공부를 열심히 하세요" can not be used solely. There should be a purpose to accomplish by studying hard. so, it can be "대학에 진학하려면, 공부를 열심히 하세요" something like that. means, "If you want to enter the university, you should study hard"

Hi ! I'm currently starting to learn Burmese. I have a long trip planned for next year and would love to speak a little bit of the language at least. I'm fluent in english, since I lived all my live in the US, and in Spanish too. So let me know if you are interested in exchanging some language knowledge. Cheers! Juan