NASA lost the original recordings from the Apollo 11 first landing on the Moon. The TV signals were beamed to Australia (to a dish in Parkes, NSW, to be precise), and then sent in reduced quality to the USA. All known copies are from the lower quality video. The original high quality ones have never really been publicly seen and NASA has lost them. This, of course, is fodder for those who think the whole thing is a Government Conspiracy.
All of the data from the receiving antennas was recorded on half-inch tape reels, so the original TV footage almost certainly survives in the archives. The problem is that the tapes were archived without reference to what kind of data they contained. Needle, meet haystack.
Not to set off the Epileptic Trees again, but you would think for something THAT important, they would have at least sent a couple of interns down to try and find it. It would seem like that would be a big priority if there was any hope of finding it.
For the minutia minded, the quality reduction was because the Eagle's video camera used an exotic slow-scan format that was utterly incompatible with network TV. The broadcast version had to be sent out by filming a screen showing the high-quality version.
Both The BBC and ITV (the only two television broadcasters in the UK at the time) somehow managed to lose their coverage of the first moon landing. All that remains are a few off-air recordings and filmed inserts. A recording of Pink Floyd performing an original composition called "Moonhead" to accompany the BBC coverage also exists in a low (but listenable) quality format and is available on various bootleg compilations.
Many sporting events from the early days of US television broadcasting are at least partially lost, including the first two Super Bowls and numerous World Series games.
NFL Films (and its predecessor) were at those Super Bowls, and footage from them is seen on various league-produced specials and DVD releases, but the entire games are not available.
In 2011, an apparently genuine copy of the CBS broadcast of Super Bowl I was found. While it was missing the halftime show and most of the third quarter, and was fast-forwarded and pixelated in spots, it's still about 90 minutes longer than anything previously seen.
One lost World Series broadcast which must be particularly galling for baseball fans is Don Larsen's perfect game at the 1956 Series between the New York Yankees and the Brooklyn Dodgers. The MLB Network reconstructed the game broadcast from existing film and audiotape, airing it as its first broadcast in 2008.
Most World Series games before the advent of recorded television the most (in)famous World Series games that will never be seen on tape include the Red Sox 1918 World Series game and of course the Chicago Cubs 1908 World Series game.
Countless early news broadcasts are long since wiped, including the first televised address from the White House (delivered by Harry Truman), and the first commercial television broadcast, chronicling the opening of the 1939 World's Fair by Franklin Roosevelt.
In truth, no signal is ever really lost. Every broadcast and transmission and reflected light image travels away from Earth at the speed of light, preserved for eternity... insofar as the inverse square law permits.
Of course, since it's impossible to move faster than the speed of light, and since the signal will eventually be so weak that it's clouded by background radiation, any signal that isn't recorded by some civilization is lost forever.
In a very loose definition of "Episode", the Imperial Faberge Eggs made for the Russian Royal Family. Of the 50 large eggs made, eight are missing and considered gone for good.
The Internet in general displays a form of this. Sometimes, files, forums or entire websites can be taken down for any number of reasons. What is lost could vary from something as small as a funny post, to an entire series of files or fond memories of a community.
One of the biggest taken downs to happen to date was the sudden takedown of Megaupload by US authorities in 2012, permanently removing access to hosted files. For many such files, there was no backup. A lesser form of this is when old websites die out, along with whatever they hosted. Frequent mirrors of data by others can help alleviate this problem.
This fate almost befell the entirety of the websites hosted by Geocities, until Oocities stepped in to host the more popular of the Geocities websites. Unfortunately for some people who hold fond memories of certain message board communities from the 1990s, those are lost forever as the hosts (such as Inside The Web, a popular choice for free message board communities) have long since bit the dust (before those communities could be archived).
Countless paleontological finds have been lost over the years. For example, the original specimens of Spinosaurus and Carcharodontosaurus were destroyed during World War II during British bombing of Munich, so until the 1990s the only things paleontologists had about these genera were some old drawings and descriptions of the original specimens (although it's worth noting that the Carcharodontosaurus holotype was less complete than the material that's been found since).
The original "Peking Man" skull fossils are very likely to be discarded as trash during one of the numerous renovations and moves at the Tianjin USMC barracks building.
The original "Java Man" fossils were lost around Singapore during World War 2. Rumors persist that they were onboard a Japanese evacuation ship that was granted safe passage by the Allies (as it delivered supplies for Allied prisoners held in Singapore on the first leg of the trip) but was sunk by mistake by a US submarine on the return trip.
In addition, due to the hit-or-miss way that fossilization occurs, and the difficulty in surviving eons of geological upheaval, and the fact that up until the 19th century or so bones were often discarded or reburied as being the bones of gods or dragons OR ground up to be used in traditional medicine, there are no doubt plenty of long-extinct species that we will never be able to find hard evidence of having ever existed.
When a building is redesigned, depending on how detailed coverage is of the old design. A good example is the California Academy of Sciences - it was completely redesigned, and even though there are panoramas on the official website, some rooms weren't covered, so they are Lost Forever.
Lots of people have a photo, home video, written document, or some other item of personal importance that somehow gets lost forever.
It is said that there are dozens of missing videos and photos of 9-11, especially victims whose cameras were so terribly destroyed by the collapse of the towers, that the data or film of whatever images they captured could not be found.
Once a person dies, all of their memories, experiences, and insights that weren't recorded are gone forever.
This happens to a lot of people who find that their old elementary or high school, say, has closed down and the building repurposed for another thing entirely. This happens a lot to people who attended non-public schools, though even public schools (e.g. those located in decaying older neighborhoods with declining populations) are subjected to this. Sure, the alum has their photos and memories to cling onto, but in a way it signifies the death of one small part of one's existence.
In a similar manner, if someone was raised in a community that constantly changes (e.g. because of massive population expansions during that time) they will often find that many of the places they frequented during their childhood will have been torn down and an empty lot or field or a completely different building stands in its place. For example, a neighborhood grocery store will have been torn down to make way for a pharmacy, a department store one went to for "special occasion" clothing will have closed its doors and a Chinese buffet opened in its place, etc. Again, this signifies the death of one small part of one's existence.
"The Basement Tapes," video recordings of the two perpetrators of the Columbine massacre, have not been released to the public. They were eventually destroyed in 2011 along with all of the remaining material and evidence from the shooting, as per retention policies (kinda, they were held slightly longer than usual). No copies of these videos exist, only a scant few clips and transcripts remain.
The video of news reporter Christine Chubbuck's on-air suicide is considered to be lost due to the master tape being sent to Chubbuck's family, who are said to have destroyed it. The incident occurred in 1974, when primitive forms of video recording existed but were not very widespread, so it is possible but unlikely that a tape of it exists somewhere, but none is currently known to.
Abe Lincoln gave an early speech condemning the spread of slavery which was either so awesome the professional reporters were mesmerized and put their pencils down or so strongly worded it was deliberately suppressed.