I suspect the answer is somewhat different from Ash's, in part because the required effects are still not well-defined.
First, what exactly is "low earth orbit"? A common definition is low enough to produce more than 11.25 orbits per day, which works out to 1269 miles altitude for a circular orbit. Let's call it 1000 miles or less just for convenience.
It's well-known that for a uniform spherical body of uniform density the surface gravity is proportional to the radius. An orbiting body (presumably spherical) at 1000 miles altitude cannot have a radius greater than 1000 miles, and if it has the same density as earth, its surface gravity will be just about 1/4 g. Then the effective gravity directly beneath it will be 0.75 g. This would noticeable, alright, but I'm not entirely sure how objectionable it would be. It would also not have a simple effect on tides. On the one hand, tidal patterns are complex and non-intuitive, as the way the water sloshes around the planet is profoundly affected by the coastal and sea-bed conformations. On the other, an LEO satellite moves fast, by definition at least 12 times faster than the tidal cycle. I'm simply not up to calculating the effect on the tides, but it wouldn't be simple.
Nor can the satellite have a simple effect on the atmosphere. The air underneath it will tend to expand upwards, but since that point is moving at about Mach 3 there won't be much in the way of a "dragging" phenomenon.
Also note that making the satellite smaller really reduces effects. Cutting the diameter by a factor of two, to 1000 miles, reduces the mass by a factor of 8, and the gravity differential as it passes overhead drops to 3%.
Of course, at these scales Roche's Limit does rear its ugly head. While it's important to realize that structural materials such as steel are far better in tension than rock is (and it's tension rather than compression that counts here), the scale of the forces involved probably makes this a minor factor. But all of that is (or can be) pretty much moot. If the satellite is built to the proper shape, and its spin adjusted to make it effectively tidal-locked from the git-go, it will already be in the minimum-energy conformation and will not be affected.
Actually, I suspect that the most important immediate effect that would trigger a response is its effect on other satellites. Even a fairly small object would grossly distort all orbital dynamics for any other LEO satellites and would make such orbits unstable. This would probably extend to destabilizing geosync satellites as well, so we'd lose both communication satellites and things like GPS.