If I missed anything feel free to add it is as a separate answer or edit this one.

Versions that introduce breakage are written in bold:

• 1.8.9
  • Unrelated bug fixes
• 1.9 Combat Update
  • New end: There's currently a bug that only allows respawning the ender dragon on portal located at (x=0,y=0). Check MC-104897
  • Brewing stands now require blaze powder (will break old auto brewers)
  • Item elevators based on collision will no longer work (other like the ones that push solid blocks into the items still do)
  • Mob spawning changes (will cause issues with farms with certain assumptions about spawning rules)
    • Mobs can spawn on upside down pistons
    • Mobs can no longer spawn on rails
  • Mobs avoid cacti and fire (might break some weird farms)
  • Mobs no longer glitch through blocks moved by pistons (breaks witch farm)
  • Other:
    • Can no longer craft enchanted golden apples
• 1.9.1
  • Hoppers can now push and pull items to and from chests that are blocked from opening.
• 1.9.2 - 1.9.4
  • Unrelated bug fixes
• 1.10 Frostburn Update
  • Endermen can now spawn in the Nether
• 1.10.1 & 1.10.2
  • Unrelated bug fixes
• 1.11 Exploration Update
  • Entity cramming maximum set to 24. If there are more entities in a single blockspace players and entities will start taking suffocation damage. Affects AFK mob farms that stack mobs (e.g. Blaze farm). Can be adjusted with the maxEntityCramming gamerule.
  • Breaks automated fishing farms using iron (trap)doors as player looking at the them will cast their line now
  • Item elevators outside of dropper based ones are broken.
• 1.11.1 & 1.11.2
  • Unrelated bug fixes
• 1.12 World of Color Update
  • No changes breaking automated farms
• 1.12.1 & 1.12.2
  • Unrelated bug fixes
• 1.13 Update Aquatic
  • Noteblocks can be pushed/pullled with pistons. Might break some designs relying on them not being pushed/pulled by a piston
  • Can now place same type of chest next to each other without them connecting
• 1.13.1
  • Squid now spawn only in rivers, oceans or beaches, breaking squid farms that are outside of those biomes (e.g. in the dessert)
• 1.13.2
  • Unrelated bug fixes
• 1.14 Village & Pillage
  • Beds replace doors as to what counts as a village and what doesn't
    • This will affect all iron golem farms.
    • Will affect villager farms
  • Trading has been overhauled (including keeping the lock on a trade until the next day). Old villagers will keep their trades though.
  • Destroyed leaves can now drop sticks (might be of importance if your tree farm features an item sorter)
  • Dispenser can now shear sheep with wool
• 1.14.1 -1.14.4
  • Unrelated bug fixes
• 1.15 Buzzy Bees
  • Dispenser can interactive with bee nest / hives (useful for farms)
  • Dispensers can now fill water bottles
• 1.15.1
  • Unrelated bug fixes
• 1.15.2
  • Oak and birch saplings grown with a flower 2 blocks away within a 5×3×5 volume centered on the sapling have a 5% chance to generate a bee nest on the side of the tree. (Might break some tree farms)
• 1.16 Nether Update
  • Redstone changes
    • Single redstone dust is now a cross powering blocks on all for sides (can be toggled with Use)
    • Redstone will provide power to blocks on sides it shows a visual connection to, and not do so on those sides without a visual connection.
    • Wire that goes over a block A can already power a block B next to A if the wire leading up to the block is power but not the one on top.
  • When fishing, treasure loot can now only be obtained by fishing in open waters. -> This will break auto fishing farms that farm for treasure loot
• 1.16.1
  • Unrelated bug fixes