Can the second "that" be omitted in the following?
My teacher tells me that there is not much time and that I should study hard.
Yes it can be omitted. In this case, it probably does not make it ambiguous to omit it, but sometimes it can do, and then it is clearer to repeat that.
Personally I would not omit the second "that". It adds to the point about time with a reference to the previous point by your teacher. It also identifies it is about your teacher's views and not your own view to study hard.
Overall Colin Fine's answer is correct as well. However Colin correctly pointed out about possible ambiguity in the sentence without the second "that". As there are three ideas in this sentence they should be clearly separated with a second "that". In this way your teacher's clause is linked with the clauses of your teachers' two subsequent and separate ideas.
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