The hotter the engine runs, the more power it makes (to a point!) for the amount of fuel going in.
The hotter the engine is, the less heat energy it can absorb from the combustion... so it helps economy by a tiny amount, and reduces emissions by a tiny amount.
I changed our E39 525i (M54) to a fixed temp stat and did away with the Map-cooling. E39's didn't bring up a check engine light for no map-cooling, but did have a fault code (Which was dumb, because when the thermostat heater burnt out, they just ran really hot ALL the time!). The 6's didn't run *that* hot on the mechanical only anyway (98?), like the V8's did (108?).
I didn't notice anything dramatic change as far as economy, but you want to make sure the new stat stays 100% closed until the engine is at operating temps, or it'll take longer to heat up and use way more fuel during the cold start enrichment phase.
My first kit had a bleed hole drilled in the thermostat and it was stupid and unnecessary. I replaced it with a solid one and it was way better.
Later cars may be a bit more complicated, but that unit looks like it still has a map-cooling heater anyway?