I have heard both variant used:
- I insist on that he does his homework now.
- I insist that he does his homework now.
Which is correct?
Is the following sentence without prepositions grammatical?
- I insist he does his homework now.
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The form "insist on" is used with nouns and noun phrases; whereas "insist" alone is used to with finite clauses (clauses that can be sentences on their own).
For the three examples that you gave, two are grammatical:
These are both okay, because "that" is usually optional when used to introduce a finite relative clause.
The other example is not:
This is not grammatical because "that he does his homework now" is not a noun phrase. As in Tᴚoɯɐuo's example, you could turn the latter part of the sentence into a noun phrase with "I insist on him doing his homework now".