In The Empire Strikes Back , we see TIE Fighters in the atmosphere of Bespin, not the vacuum of space, when the Millennium Falcon is being chased after rescuing Luke.
(See around 2.44)
As Bespin is a Gas Giant they obviously do not fly over land, but they are below a floating city known to have a breathable atmosphere.
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In The Empire Strikes Back , the Millennium Falcon hides in a "cave" on an asteroid.
The TIE bombers fly low over the asteroid dropping bombs.
I argue that the asteroid had a non-trivial atmosphere: Han and Leia explore outside the ship.
Although they carry supplemental breathable air, there is clearly enough atmospheric pressure that they do not immediately die.
Since the "cave" (actually a space slug) is open to the surface of the asteroid, it would be near-vacuum if the asteroid didn't have an atmosphere.
They seem to experience a normal gravity as they walk outside the Falcon , suggesting that the asteroid has enough gravity to hold an atmosphere.
The mynocks appear to fly using bat-like wings, which implies a certain minimal amount of atmosphere.
The space slug appears to have nostrils, which implies it breaths an atmosphere.
(I'll concede this point, though, since what appears to be a nostril may just be a shadow.)
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This is questionably canonical, but there's an easter egg in Episode 2 that includes three distant TIE fighters and X-wing during the chase seen through Coruscant at the start of the film.
Coruscant has a breathable atmosphere (evidenced by the chase scene is in an open air vehicle).
Wookieepedia says that: During the chase through Coruscant, after Padmé Amidala survives Zam Wesell's assassination attempt, three TIE fighters are seen chasing an X-wing starfighter.
They can be clearly seen inside the red rectangle in the following images, from this site.
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Edit
It seems that I was wrong :(
It's always difficult to prove a negative, but all references to TIE fighters I can find from the two movie trilogies indicate that we do not see a TIE fighter flying in an atmosphere (which has now been proven wrong)
However...
In Canon we do see this!
Wookieepedia tells us that:
They were able to operate both in space as well as in planetary atmospheres , acting as fighters or scouts.
according to the canon work Star Wars Rebels: Spark of Rebellion
We see this at 0:45 in the below clip:
But this isn't from either of the movie trilogies as requested in the question!
Nevertheless, we do know from canon that it is possible.
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This is slightly outside the scope of the question (not Original Trilogy), but in New Disney Canon - even pre-Episode-7, we do see this.
In the novel Aftermath , we see TIE Fighter being commandeered by Nora, flown in the planet's atmosphere, engeged in dogfight with other TIE Fighters, and eventually crashed into Satrap's palace.
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Tie fighter engines are ion thrusters , which wouldn't work outside the vacuum of space (i.e. in the presence of other ion atoms).
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In the trailers for  Star Wars - Episode VII:  The Force Awakens , we see TIE-type vessels flying near the ground, in the atmosphere of a planet (the planet itself looks a lot like Tatooine).       Have we ever  seen  this before?  I am specifically looking for:   Footage from the two existing film trilogies   Spacecraft from the TIE family, either fighters or bombers   Operating in full-atmospheric conditions, not the near-vacuum of space
Have we ever seen TIE fighters/bombers operating in atmosphere before?
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<star-wars><spaceship><the-force-awakens>
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As TIE-fighter engines were ion thrusters, they would not work outside the vacuum of space, and indeed it is doubtful that there are actually examples of TIE-fighters flying in full-atmospheric conditions. However, there are numerous possible examples of TIE-fighters flying in what could be considered to be atmospheric conditions. These could include when they fly in the atmosphere of Bespin or when they drop bombs on an asteroid, both scenes in The Empire Strikes Back. A number of scenes are mentioned in which TIE fighters do fly in atmospheric conditions, but some canonical sources nonetheless state that their engines would not be able to operate in such conditions.
TIE fighters fly in the atmosphere of Bespin in The Empire Strikes Back. TIE fighters fly low over an asteroid that has an atmosphere in The Empire Strikes Back. TIE fighters fly through the atmosphere of Coruscant in Episode 2. Wookiepedia tells us that TIE fighers can operate in planetary atmospheres as well as space. TIE fighters use engines that cannot operate in planetary atmospheres.