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In my humble opinion, and 45+ years of automotive "backyard" experience, plus having owned many BMW's and high performance cars, the answer is no. There may be some increased oil consumption due to higher compression and its impact on oil flow around the piston rings, however particular to the 1M, the tunes most of us have are mild relative to the increased load placed on the mechanical parts of the engine.
That being said, all engines burn oil, each at a different rate depending on how the motor was broken in. This is especially true with motors using synthetic oil, which is more likely to find its way past the rings.
Your most likely issue is a bad sending unit. You don't mention the miles, but most of our 1M's, regardless of mileage are virtually new with respect to the motor.
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