Why won't my Central Heating Boiler turn off with Nest thermostat installed?
I fixed it myself. In the manual (page 18) there is the following image.



As shown on the photo in the question:



From left to right:
The red "bridge cable" is meant to close the circuit in order for the next two (bus +/-) to work. But the bus connectors only work with with an eBus system. Which is not compatible with the Nest Heatlink. 

The two wires that are used to connect the boiler to the heat link (the left brown/blue wires) should be connected to the 4/3 connectors instead. So after removing the red bridge cable and connecting the heat link to 4/3 of the boiler, it worked fine.

Disclaimer for those with the same problem: I'm not a electrician, although I did check the solution with one. Try this at your own risk.