What would permit such a system to work?
Magic, plain and simple. What you're asking for is so far away from any plausible scientific or engineering feat there's not really any other answer. Even in scifi, the antigravity fields tend to be confined to very small volumes, so this is even more unsound than star trek or star wars.
Are there any unforeseen consequences?
Your plan is almost entirely made of unforseen consequences.
Consequence the first, for example: once the gravity beneath your satellite is neutralised, what keeps your satellite in its orbit? Without gravity, it'll shoot off on a tangent away from Earth at orbital velocity until the gravity-neutralizing effect stops working. A very powerful rocket would be needed for station keeping.
Speaking of station keeping, you'll need to have the area of effect more-or-less centred on the thing you're lifting all the way up. This means sticking your lifting satellite in geosync orbit, or having it in a forced orbit at low altitude (requiring very powerful rockets) or on some kind of dynamic orbital ring.
There will be some interesting atmospheric disruption along the zone that the antigravity system affects. I'm not entirely certain what this will look like... there will be a general upward movement as the stuff at the top escapes into space and the high pressure stuff further down tries to equalise with the lower pressure stuff above it and air will be sucked in from the surroundings at lower level and blown out at higher levels, possibly forming some kind of large toroidal vortex that might actually form a tornado. It'll also be subject to interesting coriolis effects if you're not pointing an antigravity beam along the Earth's axis of rotation, but this isn't a large effect (see this answer of mine for some discussion of coriolis effects for things suddenly not subject to gravity).
What happens to the ground (or ocean) in the volume of effect is probably a bit more dramatic and destructive. I think the ocean will fountain up, pushed by pressure from below with substantially more force than the air, because there's potentially a lot more ocean pushing down and providing pressure from below. I haven't the faintest idea if the tensile strength of the crust and upper mantle, and the viscosity of the mantle in general is enough to prevent the formation of a giant magma fountain but, y'know, it might not be.
The biggest issue is the whole "conservation of energy" thing. If you can just conjure energy from nowhere by lifting stuff up magically for free, then turning the gravity back on, then you've broken huge amounts of physics, and I can't help you. Clearly, if you can create free energy then merely using this thing for spacelaunch and keeping your spacecraft comfy seems a bit silly... you should be blasting yourself up the Kardashev scale.
(speaking of breaking physics, what happens if you use one of these things on a black hole? can you now cross back through an event horizon without waiting for the black hole to evaporate?)
If you can't create free energy, then you'll have to pay for every single joule of gravitational potential energy you're imbuing the material you're lifting... this would be about 3.4 MJ/kg for lifting something from the surface to the altitude of the ISS. Obviously lifting any useful amount of mass requires a large amount of energy, and the mass of a column with cross-sectional area 1m2 reaching all the way down to the Earth's core would be gigantic and need such a powerful energy source as to effectively be unfeasible.
Clearly you need some kind of additional constraints on your AG system... it nullifies gravity between two plates, perhaps. Then you can stick one of the plates in orbit and one on the ground, and only have to lift the atmosphere (10 tonnes per square metre at the surface) plus the weight of your spacecraft. If you were clever, you could build a dynamically suspended tube (maybe using a space fountain, or maybe the antigravity system could do this trick too) so as to prevent lateral air movement during operation, preventing the formation of a tornado.
Suddenly though, if you can make a confined gravity nullification system then maybe you can simply make a box that renders its contents effectively massless (though you might wanna consider how inertia works when moving something confined in this way, because I'm not gonna touch that problem here) and just attach that to a regular spacelaunch system. Sure, you can't lift football-field sized things into orbit, but on the other hand you can do all sorts of other neat things that don't break the local atmosphere or crust. You have now more or less re-invented cavorite, or as was pointed out by DWKraus, the spindizzy, or any number of their descendents.