The that can be omitted from a relative clause except when the relativised noun phrase functions as the subject of the relative clause:
He was the person I saw/that I saw
He was the person that saw me.
*He was the person saw me is not grammatical in standard Englishes, though it is possible in some dialects.
So in your example
Select the exercises the course contains.
would be possible, because the exercises are the object of "contains" in the relative clause; but not *Select the course contains the exercise.
It is nothing to do with the tense, and in fact your question has nothing whatever to do with tense. It's true that one of the uses of the -ing participle is to create a continuous tense, but only when it follows a part of the auxiliary be. Here there is no such auxiliary, and no continuous tense: only a participial clause introduced by containing.
When you use this participial clause it is replacing the relative clause, so you cannot use the relativiser that (or a relative pronoun such as which).