It takes a pretty special QB to make it despite being 6 feet or less. Has to have either great anticipation or mobility or both. There are plenty of good examples of superb college QBs in that height range. And if a player is listed at 6 feet, he's less.
The reason given for ideal height being 6 foot 4 + is better sight lines and ability to throw shorter passes on a flatter and faster trajectory.
A large human head is about 9-10 inches from top to chin. The eyes are about halfway down, so in an NFLer 5 inches. The shoulders/trap muscles begin to extend laterally at about chin height and fall inferiorly with increasing distance from the spine.
So if a 6 foot 4 QBs eyes are at roughly the 5 foot 11 level that means to see over a lineman's head that lineman has to be 5 foot 11 or less. Of course offensive linemen bend their knees this dropping their total height about 3-4 inches. So they can now be 6 foot 3 or so. But, the QB also has a slight knee bend so that probably puts the upper limit at 6 foot 1.
Obviously there are no linemen that height, therefore no QB, except maybe Osweiler can see over a lineman's head. However, there are still the traps/shoulders/backs to see over. If the average lineman is 6 foot 5 and their knee bend drops them to 6 foot 1 and their traps begin at approximately the 5 foot 1 level that means a QBs eyes have to be at the same level or higher. As the eyes are roughly 4-5 inches from the top of the head that means a minimum QB height is 5 foot 6. Hmm. Seems like a shorter QB should be able to see over the shoulders.
However, there is still the issue of batted passes, and arm strength. In general, those d lineman are going to be anticipating passes and waiting for that opportunity to knock it down. Simple physics here. Same with arm strength.
Still, a QB that's elite level at anticipating where receivers will be, reading defenses, accurate while rolling out etc should be able to play, no question. Brees, Russell; in college Kellen Moore, etc