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One day, the earth suddenly gets covered by a spherical barrier that allows waves (light from the sun, radio, etc.) to pass both ways, but not matter. Let's ignore the feasibility of that kind of barrier existing and say that it's just there now. Anything with mass bounces right off as if it were an impenetrable wall. The barrier hovers at around 200km from the surface.
Will there be a noticeable change to atmospheric conditions on a timescale of around 1000 years?
Absolutely nothing.
From Wikipedia:
Atmospheric escape of hydrogen on Earth is due to charge exchange escape (~60–90%), Jeans escape (~10–40%), and polar wind escape (~10–15%), currently losing about 3 kg/s of hydrogen. The Earth additionally loses approximately 50 g/s of helium primarily through polar wind escape. Escape of other atmospheric constituents is much smaller.
Some math:
The atmosphere would gain 0.000001837% mass.
Completely negligible.
The atmosphere weighs $\pu{5.148 \times 10^{18} kg}$.
Earth loses $\pu{3050 \frac{g}{s}}$ or about $\pu{96.24\times 10^{6}\frac{kg}{year}}$. Or about a fraction in the ballpark of ${2}\times10^{-11}$ of its mass is lost per year. In other words: nigh nothing compared to the mass of the atmosphere is lost.
Stopping that neglectible loss will have neglectible effect. To even be noticeable as a 1% difference from the predictions, earth will need to run on about $10^9$ years. Remember that earth is only about $4.5\times10^9$ years old.
200 km is well below the common Low Earth Orbit, and thus almost no new satellites can be put up, especially nothing geostationary or with sensible retention times.
The ISS is at a 413-422 km orbit, and thus will be cut off.
Meteorite impacts and debris will stop at the 200 km line, changing the pleiads and others from straight shooting-stars into ones with a bend.
Now, there is one thing that we need to ignore for the above to be true: Relativistic mass. As long as we ignore that light can behave like a particle that has a mass, everything is hunky-dory and earth is fine.
The moment that the shield does not ignore relativistic mass, things get bonkers: the shield turns the Albedo of the planet from something like 0.3 to 1. That means not 30% of the light falling in is reflected to the stars, 100% are. That means multiple things:
On the day-to-day of things, you'd not notice much difference, but anytime a major volcano erupts somewhere in the world, the climate impact it has would be increased. For example, the recent Tonga volcano eruption is reported to have sent about 8% of its mass ejection off into space. This % only goes up as a volcano becomes more powerful; so, the bigger the volcano, the bigger difference this shield will make. For example, the Mount Tambora eruption in 1815 was about 15-20 times as powerful as the Tonga eruption. As it was, this eruption caused the "year without a summer", but had there been a shield like this, it could have possibly doubled the intensity of the volcanic winter.
In the past 1000 years, there have been 8 eruptions on Earth big enough to cause noticeable climate impacts that would have been made at least 10% worse by this shield. So, most of your next 1000 years, would be unaffected but within that time period, you'd have a few bad years where it makes a significant difference, and probably at least 1-3 where things would get really really bad for anyone unfortunate enough to be alive at the time.
There would be very little effect upon Earth itself as others have pointed out. All existing satellites would eventually cease to function and would not be able to be replaced so no more space exploration and GPS, Satellite telecommunications, Earth observation and more would be ended or very seriously disrupted.
On a more positive note there would (presumably) be no danger of a civilization destroying asteroid impact.
The escape layer of the atmosphere is several thousand kilometers up, which is many times as high as the shield at about 200 km.
So the atmosphere above the shield would continue to slowly escape into space, but the shield would stop gas from lower down from moving up to the upper atmosphere. Thus the atmosphere above the shield would diminish faster than it does now. However, the slow rate of atmospheric loss should mean that will not be a problem in the long range, and certainly not during the time scale of most stories.
Micrometeoroids burn up lower than the shield, so people would no longer see meteor trails in the night sky. Instead space dust would hit the shield and be shattered or even vaporized by the impact, or possibly merely sit on the shield. Larger space rocks and asteroids would also be stopped by the shield, perhaps exploding spectacularly.
Since dust particles are opaque, the higher the percentage of the shield's surface which was covered by dust particles, the lower the percentage of the sunlight hitting the shield that would reach the surface of the Earth. I expect that dust buildup on the shield would probably be far too slow to be an important factor in the story.
The ionosphere (/aɪˈɒnəˌsfɪər/)[1][2] is the ionized part of the upper atmosphere of Earth, from about 48 km (30 mi) to 965 km (600 mi) above sea level,[3] a region that includes the thermosphere and parts of the mesosphere and exosphere. The ionosphere is ionized by solar radiation. It plays an important role in atmospheric electricity and forms the inner edge of the magnetosphere. It has practical importance because, among other functions, it influences radio propagation to distant places on Earth.[4] It also affects GPS signals that travel through this layer.
https://en.wikipedia.org/wiki/Ionosphere
So about a fifth of the Ionosphere would be inside the shell and about 4 fifths would outside the shell. If being outside the shell caused major changes in the Ionosphere it could affect long distance radio on Earth. But I don't think changes in the ionosphere would happen fast enough to affect the story, unless you are planning to write a multi billion year future history like Stapledon's Last and First Men.
So far I have not been able to think of any changes to the atmosphere that the shell at 200 kilometers would make happen fast enough for the purposes of a story. But if an expert in atmospheric physics can think of a change which would happen fast enough to be important for a story, they should mention it in their answer.
