Like Don Bluth's previous 1988 film, The Land Before Time (though The Land Before Time has much more cut content), the 1989 animated film All Dogs Go To Heaven had two very dark scenes that were cut during initial test screenings; one involving the main character Charlie being hit by the car (in the general release, only the aftermath is seen), the second being a longer sequence of his nightmare of the Eclipse. The original version also contained minor cursing, which was also edited out; the vinyl soundtrack of the song Let Me Be Surprised had Charlie saying "damn," which is clearly animated in the movie.
The original uncut version was stolen, and Goldcrest Films destroyed the remaining uncut print to avoid storage fees. Gary Goldman, the co-director and producer, was interviewed about several Don Bluth movies; when asked if there'd ever be a director's cut, he is quoted as saying,
"No. The footage that was cut, was discarded, not saved. The drawings were not archived. Goldcrest Film and Television actually refused to store the original art (drawings) and, I believe most of the cells and backgrounds were destroyed or taken to the dump in the UK. 'They said that it cost them “too much” for adequate temperature and humidity-controlled storage.' 
There have been rumors of Don Bluth owning the only uncut version in existence. Unless this is true it is highly unlikely that there will ever be a full, uncut version.
Extended Eclipse Scene
Update 08/12/16: A low quality version of the uncut eclipse scene was discovered and uploaded by Tumblr user SteamRunner (the audio is bad, but the full scene is visible): http://steam-runner.tumblr.com/post/146385150575/so-i-am-taking-some-courses-in-a-summer-art
This scene was originally going to be a bit longer, but was cut to avoid a PG rating. Originally, there was a one or two second close-up of the skeleton demon that was cut before it hissed at Charlie.
But the second cut was much bigger and it involved the Darkhound. Right after it forms, it flies down towards Charlie (like in the cel at the top left) and the angle changes to the front of its face before it says: "Now, you are MINE!" before then throwing its head back laughing. The scene then plays as normal.
The Darkhound had only spoken in the cut version once, near the end of the eclipse scene.
At least one further deleted shot from the eclipse sequence that would have included a close-up of the skeleton demon was cut from the final version of the scene and was not present in the recovered footage uploaded by SteamRunner. This brief shot likely would have occurred after Charlie howls in pain due to the demons biting him and directly before the shot of Charlie jumping in reaction to something offscreen that has clearly frightened him. This missing shot's existence was verified after several original production cels from the scene surfaced online.
A rumored extended version of the scene had Charlie noticing the car just before it hit him, instead of only showing the the car plunging off the end of the pier.
As of October 2nd, 2020 a supposed still of the extended scene found was debunked as being a part of a different scene from the movie. During the showdown scene while Charlie rescues Anne-Marie. Carface's henchmen tied him up and one of them bit Charlie's foot resulting to his reaction. The uploader apparently used some sort of paint software to add their own touches to the false proof for unknown reasons. https://lostmediaarchive.fandom.com/wiki/File:Debunkingadgthcardeath.jpg
On October 24th, 2020 it has been confirmed on the official Facebook of Don Bluth by the producer of All Dogs Go To Heaven, Gary Goldman that the extended scene where we see Charlie gets hit by the car does not exist.