Why, and what would happen if you tried it?
With the way Harold White explained it in his paper the field requires some form of input to offset it, this is why in SpaceEngine now it works as a boosting factor. The ships velocity is what directly changes the field layout. Your speed plus the field strength will be your effective velocity. If you flew such a sensitive field through a non vacuum, like a planets atmosphere, it would most likely collapse due to the high amounts of positive pressure vs negative.
SpaceEngineer and I have discussed a mechanic regarding nebula. The warp fields may be slightly diminished in their effectiveness when traveling through but that is a far off in the future thing that may or may not ever be added.
Here are some of the slides I uploaded from White's paper back in 2014 which semi explain the idea behind it
If we assume you flew into a planet at warp, so the field was on before entering the atmosphere, the energy exchanged between the field and the planet equalizes to near 0 and the ship slams into the planet imparting only the kinetic energy of it's base velocity. The key bit of info here is the ship is not actually moving at FTL speeds and the energy within the field is minimal at best.
If the warp field can't be maintained, wouldn't that be more of an effect of the planet's gravity rather than its atmosphere?
Not really, the warp field is very small relative to the size of a planet. Can't really say if in real life that would be an issue, you would need to ask
Watsisname, but as far as I know from reading through the papers and asking questions on things I didn't quite understand it wouldn't be an issue.