I checked the setting to enable multiple levels and I get the following on launch.
Failed to load mods: furnace-recipe/data.lua:1:/data.lua:1__furnace-recipe/prototypes/electric-multi-furnace.lua:133:attempt to index field 'advanced-material-processing-3' (a nil value)

I got the issue. I missed that the original game does not have Advanced Material Processing 3 which I used as unlocker for the Multi Furnaces. I was lazy and just attached my mod to existing technologies and now the day came, when it came back on me. :)
The solutions is my original idea: Introduce an own researchable technology.
Already crafted or placed Multi Furnaces won't vanish, but you cannot craft new ones until you researched them.

Awesome, up and running. Thanks! :D

I got this message after updating the mod to 0.5

Failed to load mods: Error in assignID, recipe with name 'electric-multi-furnace-2' does not exist.
Source: electric-multi-furnace-2 (technology).
Mods to be disabled:
- furnace-recipe

I installed the 0.4 version and got the same message so it must be some other mod crashing with this one. The problem is, I don't know wich mods ware updated. I know there ware 3. Is there a way to check?

Got it. Probably a problem, I've introduced with the technology addition because the mod tries to add the technology assuming that the items exist when you enable the option for multiple levels.

I'll verify my guess and upload a fixed version soon.

Update: I was able to reproduce the issue. I guess in the Mod Settings menu, you've disabled the Own Crafting Branch.
I'll add a bit more dynamic under the hood to cover this issue.

Okay got it. I had to mess around with some weird behaviour of Factorio regarding my latest changes to get a smooth transition between version 0.5.0 and the new 0.5.1. For an unknown reason the crafting recipes vanished after my changes and I had to add some save migration script. Confusing stuff.

Please test, if it works for you. Otherwise just tell me.

It works now.

Thank you very much.