If you want to forget something, you basically cannot -- at least not quickly. There is a possibility that if you put it to the back of your mind, figuratively of course, for long enough that your subconscious will stop recalling it very frequently at all, and eventually the hippocampus may decide to overwrite that memory during REM sleep one night. Hypnosis may be helpful in getting this process started.
If you want someone else to forget something, the same process must be undergone. In likelihood, they will eventually stop thinking about whatever it is. But if you want this to change quickly, then find a tactful way to redirect conversation any time the event is mentioned, such that it is not at all obvious. Neural Linguistic Programming can be your friend here. Down-play its importance to the point of irrelevance, and it will be mentioned less. Eventually, some new shiny thing will catch attention, and can be an easy thing to use as a focus just in case. Again, the memory of whatever it is must be out of the conscious mind for a long time, and made so unimportant to the subconscious that it is overwritten during REM sleep, by much more important information.
When a memory is overwritten, it is not so much like a computer disk. It is more that the memory is compressed. Details are lost. This can be used to an advantage, if there is any way to implant a false version of the memory, bit by bit. Eventually, the details are reduced to mere tropes, the way our minds can remember every face we pass, even subliminally, as tropes without recalling the actual face.
At some point, if things are done successfully, the tropes will all be falsehoods, based on imagining an implanted memory. These false memories will be based upon other tropes, which makes them believable enough, but also makes them more generic the more false the memory becomes. If the tropes of the false memory are so reduced that the hippocampus realizes that they are duplicates, it may reduce whatever remains of the memory to a string of instructions (basically) to recall the like tropes from their other storage location, making the memory yet less concrete, and more open to suggestion, should it ever be brought up again.
This process would need to be dedicatedly undergone with every person who witnessed whatever the event was, and from their own perspective, to avoid a more accurate recollection in the eventual future.
Perhaps you could take a shorter route, and wait it out. Things pass. Forgiveness for wrongs comes, or not. Embarrassing moments become less important as time goes on.
Thank you very much for this advice. I suppose I could try something like hypnosis. I know I can't make it instantly disappear, but I needed something to aid me, since anything else I had tried had failed.