I suspect that at best you could get partial cancellation as long as the mics don't move at all. How about having the director not speak during takes? Also, it's standard practice to use a boom mic to capture the dialog. A mic on the camera isn't likely to give the best results. Bigger productions do ADR, additional dialog recording, where the actors rerecord their dialog after the fact while watching the scene. In that case background sound is captured on the set or some other way to make it sound real.

We are shooting with a 180 degree VR camera so the boom mic is out. With my actors adr will be a pain. As it is right now I was expecting to go in and cut out the directors voice and fill those spots in with background noise, but I was hoping there was some sort of way to highlight a section of affected tracks, and then have the director track used as a source for canceling.

In the best world, the director shouldn't be talking while recording... Comments and instructions should be made off camera while actors can give him/her full attention. Then when the recording starts they would concentrate on acting not listening to someone...
Having mics on the camera gimble is good to grab ambiant sound and noises, if that's what you are looking for, it comes with the downside of picking everything, talks, cars, planes etc...
Big production will record ambiant noises when no one is around then have lavalier on actors OR will do post sync the voices in studio.

Playing with phases and trying to cancel a vocal from 2 differents mics with different frequency response and different distances will be hard.
Maybe a product like this could do an ok job :
https://www.izotope.com/en/products/repair-and-edit/rx/features-and-comparison/de-bleed.html

This is like the old question of "How do you un-bake a cake?"

The problem is that, although there may be tools and techniques that could reduce the amplitude of a particular form of interference (the director's voice in this case), any technique that is going to produce a real degree of attenuation of that source will also affect the sound quality of the intended sources, since they are all speech. It's not like separating speech from music, where there are ways of identifying the characteristics similar to those that our human hearing mechanisms use to separate these different types of sounds.

In addition to the algorithmic problems, the vocal signal from the director's headset mic will be some tens of milliseonds ahead of however much of it is picked up by the actors' microphones, and this time will vary as the director moves his head or the camera pans. It could be that you may find that camera-mounted microphones do not produce an acceptable soundtrack recording in any case, external interference or not.

Overall, it's not looking good for getting a director-free soundtrack without affecting the quality of the actors' voices. Personally, I just can't imagine a director who would talk audibly during a shoot.

I agree wholeheartedly with my friends here, but to add one more thing to the conversation - some Sony camcorders have an audio setting called "My Voice Cancelling" that theoretically suppresses the voice of the camera operator. I haven't heard it and cannot say whether it works flawlessly, has some artifacts, or is totally useless, or how much it suppresses the voice. In any case, I'd be reasonably sure it has 2 forward facing mics and 2 rear facing mics all in extremely close proximity. And any "cancelling" in that setting is achieved due to the camera operator's voice being virtually equidistant and mere inches from all 4 mic capsules. If that's the case, it would only be able to suppress the director's voice if he were operating the camera. Put him any distance away from the camera and off axis and (in my opinion) the limited effectiveness of all of that goes out the window.

Quoted Features from Sony site:
Sony’s new My Voice Cancelling (available on select models) suppresses unwanted vocal frequencies from immediately behind the camera, without affecting the volume of the subject’s voice and other foreground sounds.

You'd probably have better luck hiding any of the tiny wireless mics available on the market in their hair or costumes.

Then again, to the point that has been hammered home already, a director shouldn't need to bark instruction during the take. If it's for cues or choreography find another inaudible way to signal the changes - or cut to another camera, or B-roll or something, and get creative dovetailing them together.

And you'll pardon me, but if it's a porno where the "actors" need constant coaching just turn up the music bed and overdub some "vocals" and please disregard the suggestion of hiding tiny mics anywhere on their person. :)

Perhaps not really for here, but back in the late seventies when I first started this lark, one of my first audio jobs was to create the audio for these kinds of movie. All the actual films were at that time silent 16mm, shot in Holland, and when video recorders suddenly became common, there was a market. So the film was telecine transferred and then a group of actors and techs had to record the totally missing audio. It was pretty simple to do, with Foley type er, noises, and the actors did the voice noises. The result was excellent, and it paid well - the difficulty was somebody spoiling the take by laughing - which happened all the time. This is, by the way, and excellent adult party game - who can go the longest without laughing - two teams and a 4 minute clip. 20 seconds is doing well.