Can you run a 6s battery with the DJI FPV air unit?

Yes, absolutely. But make sure you are using power from a 9V regulator on your flight controller.  There are many flight controllers (like the Holybro Kakutef7 HDV) that are designed specifically to power the air unit.

For the goggles, use a 4S pack, they can't handle any more than that.

OK thanks, I'm ok then, and have 4s for the goggles

Actually, I've discovered accidentally that goggles work perfectly fine plugged into a 6s pack all day long. Pretty sure I've seen someone else do that and even document it in a video, will try to find it.
(I wouldn't dare trying 6s with the Air Unit as I've had spikes on 4s vbat kill one, just go with Vista if you want a 6S build.)

Here you go, at the start of this video KababFPV explains how we accidentally ran 6S on his DJI FPV goggles:

No.  6S is absolutely fine on a properly built DJI drone with the right 9v regulator.  I've been flying the same rig on 6S every day for a year now.   And if you want to send your goggles to an early grave running 6S, go for it.   There's a reason their engineers rated them for 4S max.  

mc54 Posted at 11-25 21:32
No.  6S is absolutely fine on a properly built DJI drone with the right 9v regulator.  I've been flying the same rig on 6S every day for a year now.   And if you want to send your goggles to an early grave running 6S, go for it.   There's a reason their engineers rated them for 4S max.

Thanks, that's helpful