Depends on what the owner wants. The ones that are rolling coal are removing all the filters then deliberately running their engine rich. The black smoke is literally unburnt diesel. Obviously when you're spraying huge clouds of diesel, your fuel efficiency is going to go down.
After Dieselgate, Volkswagen brought their diesels into compliance by software patch. People did dyno runs of all VWs before and after the patch, and in many cases (e.g. the 180HP Passat) got something like 10% less horsepower and drastically worse torque curves (max torque at 1900 rpm instead of at 1500 rpm) after the patch. Fuel economy also went down by around 10% because of the patch.
Makes sense from a thermodynamic perspective: your engine runs with more power/torque and more efficient if you run it hotter. But running a diesel hotter makes much more NOx.
Some parts of the emissions control systems (I was informed by competent technicians, but not yet adequately "educated") work through burning extra gasoline and fuel - they are active, they use energy.
The extra intake may seem pretty dramatic - though of course full data for comparison should be in order... It should be possible to compare euro-4 vs euro-5 model consumption through spritmonitor.de (through selecting by year) - I have no time right now but may do it later, and possibly report.
no, it's the opposite. Emission controls result in worse fuel efficiency.