The resulting item is usually an overlap of both the original and new icon, with a name that may be the new item or another mishmash (e.g., "Irish Breakfast Tea Wine" or "Magnolia Sapling Honey"). The items won't damage your save, but they (usually) won't work properly either, so the easiest way to handle them is simply to ship them like any other item to be sold. Sell prices reflect the new item, so if you were lucky enough to have, say, your Octofruit turned into an expensive legendary wine, be prepared for quite the payout! The most likely items to experience the shuffle are flowers that come in multiple colors or items that are in a machine (e.g., a keg or custom machine) - if you empty out your machines and sell any colored flowers before adding or removing JA mods, you'll likely be safe.
Rarely, an item will be glitched enough that it will cause SMAPI to close out of the menu to prevent the game from crashing. Should this happen, the best way to correct is to close the game without saving, remove/add back any new mod(s) that added/removed items, and sell any items that would be affected by the shuffle before adding/removing back the mods you want to add.
To ensure you avoid the Json Shuffle, choose which JA mods you want to use for a playthrough at the start of your game, download and install them, and then do NOT add or remove any for the rest of your playthrough. Updates to mods you're using may contain new items, so keep that in mind and read the changelogs before updating JA mods. If you do add or remove a JA mod during a playthrough, be prepared for shuffled items. You can use a mod like CJB Item Spawner to restore your lost items.
Dynamic Game Assets (DGA), a recently release by Spacechase0 that is effectively JA 2.0, does not experience shuffle as its way of coding items is different, however most mods have not been updated to it as it is currently only compatible with Stardew Valley version 1.5.5 and lacks compatibility with other JA-using frameworks (e.g., Producer Frameworks Mod. ConcernedApe has also looked into the possibility of changing item coding in 1.5.6 to fix JSON shuffle on the base game level.
Custom Furniture has its own version of Json Shuffle: if you remove a furniture mod whose furniture you were using, the removed items will turn into other CF items.
There is also a related shuffle for NPCs - if you add or remove NPCs during Winter, it can cause your secret giftee for the Feast of the Winter Star to change.