I understand your concern, it's because some native speakers skip certain words while having a normal, fast conversation with others and yes, they are grammatically incorrect.
The sentence "We'll have to leave September 1st" may sound as if you have heard it somewhere, but still it's grammatically wrong. Date, time and venue require a preposition if used after a verb. But in many informal, short conversation people skip these grammar rules, but that doesn't make it correct.
Therefore, the sentence "We'll have to leave on September 1st" is grammatically correct.
Sometimes, native speakers are just lazy ;)