Learn English – Insist+ that-clause

There's something about this construction that is unintuitive, possibly the way that I initially parsed it. Syntactically, this works. The reason for it is that the verb phrases fork from requires. For instance:

Additionally, to achieve goals, the organisation requires that:

• it have hardworking leaders;
• people get motivated;
• attempts be organised;
  ... etc

The subjunctive, as TRomano noted, is the form it uses. In answer to your question, you can use either form, though anecdotally the subjective is falling out of use. Some verbs are more felicitous with the subjunctive, though, as in your example:

• I recommend you go home and wait until more is known of the issue
• I recommend you to go home and (to) wait until more is known of the issue

However, require does not have this issue:

• I require you to work hard.
• I require that you work hard.

That is, you can use either.

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:

• I insist that he does his homework now.
• I insist he does his homework now.

These are both okay, because "that" is usually optional when used to introduce a finite relative clause.

The other example is not:

• I insist on that he does his homework now.

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".