This thread is a work-in-progress. If you encounter any other missables I've failed to mention, please let me know. Be sure to keep alternate save files at various points, just in case.
Weapons
Apocalypse (Arubboth/World of Darkness): The corresponding chest to Masamune in Pandaemonium, it can also be stolen from Marilith in the World of Darkness. This is how you should obtain it, because you can't fight Garland otherwise and can still encounter Chaos by using Noah's Lute in one of the crystal rooms within the World of Darkness.
Auto Crossbow/Chainsaw (Fynn): These are technically not missable, however if you don't get them in Fynn, you'll have to get them as rare drops (or steals) from Death Machines in the Tower of Owen. The Chainsaw chest contains a Limit Ring (also obtainable elsewhere) in Act III, and the Auto Crossbow becomes a Minerva Bustier (which again, you can get another way).
Diamond Mace (Jade Passage/Raqia): Either obtained from the chest encounter with Scylla in Jade, or stolen from Ashmodai in Raqia. Scylla's Holy Rod can later be obtained on the Floating Continent.
Gungnir (Castle Deist): Very rarely dropped by (or stolen from) Odin.
Holy Lance (Jade Passage/Pandaemonium): You can obtain this from a chest in Jade or by stealing it from Sandy, but if you choose the corresponding chest in Raqia and don't steal it from Sandy, it's gone for good.
Longinus (Floating Continent): Stolen from Zurvan on the Floating Continent.
Lust Dagger (Jade Passage): In a chest guarded by Ogopogo on B2. The corresponding Moonring Blade, guarded by Fenrir in Raqia, can rarely be dropped by Kunoichis in the World of Darkness. You can also steal the Lust Dagger from Fenrir and get Ogopogo's Murasame on the Floating Continent instead.
Masamune (Pandaemonium): This can be obtained in three ways: the chest in Pandaemonium that contains an encounter with Garland, by stealing it from Chaos in either Arubboth or the World of Darkness, or by stealing it from Perfect Chaos in the post-game. You should open the chest and fight Garland in Pandaemonium, since it's the only way to ever encounter him.
Masamune 100 (Floating Continent): Stolen from Sefilos on the Floating Continent.
Sage's Staff (Arubboth/World of Darkness): Corresponds to the Diamond Mace in Jade. Can also be stolen from Lich in the World of Darkness.
Save the Queen (Floating Continent): Stolen from Sophia on the Floating Continent.
Stardust Rod (Pandaemonium): A chest in Arubboth also contains this, however if you get that you can never encounter Sandy, Cindy, and Mindy. The Stardust Rod is stolen from Mindy, but you may steal a Mist Ring instead. Once Mindy dies and is revived, the "steal" state refreshes and you can steal from her again, but it's random which item you get.
Triton Dagger (Paloom/World of Darkness): This is in an Act III Schrodinger's chest in the hidden room in the back of Paloom's weapon shop, and can also be stolen from Kraken in the World of Darkness. These are the only two methods to ever obtain it.
Ultima Weapon (Machanon/Arubboth/Statue of the Gods): This can be stolen from three bosses, and it's the only way to ever obtain it. Those bosses are Ultima Weapon (Machanon/Shrine of Ultima), Warmech (Arubboth), and Rest (Statue of the Gods).
Zodiac Spear (Tower of Owen): Don't open the Lone Wolf chest (upper-rightmost) in Paul's stash in Fynn, whether in Act II or III, otherwise the spear's chest in the tower contains the same note.
Armor/Relics
Command/Magic/Summon/Support Materia: These are stolen from Owen, the Attack/Defense Nodes, and Safety Bit in the Tower of Owen boss battle, also from the Short/Long Arm and Magic/Power in the battle with the Statue of the Gods.
Esper Locket (The Feymarch): On the upper level of the Feymarch. In Act III, this chest contains a Celestriad, but you won't ever be able to encounter Ramuh if you don't get the locket earlier.
Robe of Lords (Stolen from Ozma): Since Ozma can only be encountered once, be sure to steal this from it.
Yeshua Sandals (Arubboth): Corresponds to the chest on 7F of Pandaemonium which contains Hermes Sandals.
Items/Music
Aile's Tear (Devil's Road/Arubboth): This can be stolen from Satan or obtained on 8F of Arubboth if you don't get the corresponding chest in Pandaemonium.
9. Fiddle de Chocobo (Chocobo Ranch): Since the ranch closes during the third act, you could potentially miss this song's entry for the Music Box.
10. Mambo de Chocobo (World Map): Wild Chocobos no longer appear in Act III, so encounter one while you still can.
18. Road to the Summit (Poft): The music changes during the third act.
80. Shadows Rising (Pandaemonium): Plays only when you fight Garland, so be sure to encounter him.
Bestiary
Since "???" (Bismarck's Belly) in Paul's scenario is a one-time dungeon, be sure you have the following entries before completing it: 108. Hermit, 109. Nautiloid, 110. Electrofish, 111. Water Scorpion, 112. Plancti, 113. Octokraken, 114. Sucker, and 115. Corbet
When Castle Fynn is shrouded in darkness, there are enemies encountered that disappear afterwards: 117. Griffin, 118. Centaur, 119. Chimera, 120. Iron Golem, 121. Sorcerer, 122. Dark Sage, 123. Ghost Knight, and 124. Objet d'Art
Since the overworld enemies change during the third act, be sure you have all entries up to 169. Yellow Jelly before crossing over to Act III.
170. Ramuh (Bafsk Sewers, from the enclosed mountain valley entrance) is permanently missable if you don't obtain the Esper Locket in the Feymarch before Act III.
314. Garland (Pandaemonium)
316. Sandy, 317. Cindy, and 318. Mindy (Pandaemonium)
If you miss 319 and 320 (Gilgamesh), his chest respawns in the post-game, so he can be encountered again with a different party if you wish.
363. Chocobo (World Map): Again, these disappear in the third act, so be sure to encounter at least one beforehand.