Depends on the addon and what APIs ZOS has changed. If it's not throwing up errors all the time it's probably okay, but buyer beware and all that.

Without looking at the code for a specific addon it's impossible to say exactly what one will do. If it's any consolation I use a number of outdated addons and have never had anything untoward happen except the occasional error dialog pop up.

I doubt it could cause any permanent problems directly. But, indirectly it could if you trust what the add on is telling you and it's wrong. Like say an addon that tracks trait research not indicating something is unknown anymore so you decon it by mistake. That did happen to me a long time ago.

The reason an addon can't directly cause problems is no addon can change the game data, only how it is presented to you. In other words, all Addons are UI Mods that just change what is shown to you, or how it is shown to you, or automates some processes like making multiple Glyphs for example.

Pay attention to any Addon Errors as they usually tell you what addon is causing them. Either update that addon, change it, or ignore as you see fit. But, if your staring a Nirned item and wondering you need to research it, best to go to the crafting table and look at that interface than any addon before you decon or sell it.

An outdated addon can work without problem, show lua errors due to the changes in API/data, or give wrong info because of evolution of game.

It also denotes that its author is less focused in its dev and maybe left.