For verbs like demand/contend/doubted/insisted/mandate, etc...

I know the form is Demand + That + Subjunctive

Ex: I demand that you be here on time.

But I was under the impression sometimes you could omit "that" and use the infinitive form?

Ex: I require the students to be on time.


Thus, Is "I demand you to be here on time" gramatically correct?

yes, you can do that with 'require'.
no, you can't do that with 'demand'. you also can't do it with 'stipulate', 'request', 'propose', or most of the other verbs that take the command subjunctive.

in addition, there are verbs that only take the infinitive construction (i.e., that don't take the command subjunctive), even though they are similar in meaning to the verbs that do take the command subjunctive. one such verb is 'compel': the court compelled the defendant to pay heavy fines.
you can't say 'the court compelled that the defendant pay...' because, well, you just can't.

that's idiomatic usage for you; memorize, memorize, memorize. that's the name of the game.

unfortunately, the only way to learn all the distinctions is practice, memorize, practice, memorize, practice (the vast majority of native english speakers don't fare much better with the command subjunctive, by the way; it's a fairly reliable shibboleth of the educated).

also, one more comment (in case you aren't depressed enough after the last post):

one thing you should never do is assume that, because two verbs have similar meanings, they must also have similar idiomatic usage. that's a surefire way to get in big trouble really fast.

english verbs, in terms of idiomatic usage, are unique little snowflakes; you can sometimes project the idiomatic usage of one verb onto similar verbs - this is a decent last-ditch desperate guessing strategy if you just don't know - but you'll often be wrong in doing so.