Most obviously: buyable objects. Chairs, sofas, kitchen appliances and so on. They will simply not be there, and after the house is saved, the bit of code that states they ought to be there is also gone. This can happen if, say, the object IFF is moved to a different directory, or it's simply not on harddisk anywhere because the family was downloaded from the 'net. All downloadable families I've seen come with "wish lists" of objects the player should have before loading that family.
Ditto, floors and walls. If a missing object message is followed by a house showing grass and bare wall, it's obvious what went missing. (Sadly, the game doesn't produce a list of missing walls, floors and objects. There are utilities which do that, notably EliSims, but this program doesn't look inside FAR files and so reports standard Sims files as missing!)
Skins: here, at least, a name is given as a clue of what skin may be missing. The skins check is applied not only to the members of the loaded family, but also to any Sims visiting. A new outfit is put in the bodystrings, and the old one saved behind a semicolon. If the right skin files are supplied, the Sim will go back to his/her right outfit. (Buyables may also have skins, but I don't know if, say, a missing wall phone texture would be reported as missing skin or missing object.)
Incidentally, a skin is also reported missing when the skin itself is there, but the CMX that goes with it is missing or wrongly named internally. Meshes are never reported missing, they either show up or they don't.
Not-so-visible buyables: a custom phone plugin, for instance. I'd tried out a plugin, disliked it, and deleted it. The house where I'd used it then kept reporting missing objects. So I sold the telephone and bought a new one. No more messages.
Sims! At one time I was experimenting with renaming Sims using a hex editor. The house IFF stores family user file numbers, but also first names. If a Sim with that first name can't be found, this is seen as a missing object, although the family member itself will of course be present.
(House IFFs also store skin file names. I've had a house complain of a missing texture that I knew none of its inhabitants were wearing, and discovered it belonged to a visiting Sim. I've also made Sims to test certain skins, moved them to a house, evicted them, deleted them, deleted the skins, sent new Sims into the abandoned house, and got complaints of all the skins that were supposed to be missing! Apparently eviction doesn't clear out all traces of the ex-resident!)
Invisible, release-specific objects, like pending social interactions.
Each time I copy a Deluxe family to a Livin' Large environment, I get a
"missing objects" message. No physical object is missing. Everything
which ought to be there, is there. Saving the family once gets rid of
the message. It's annoying, but harmless.