Do mods disable achievments?

Everything that is activated from within the Mod-Menu at least.
There are some workarounds, such as dlc-mods and just direct file-edits.

There are also some mod managers that allow you to enable (many, not all) mods before the game starts and thus don't turn off the Achievements - JdH's CiV MP Mod Manager for example can do exactly that, although its main function is to allow multiplayer matches with mods.
 
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