The Navy even had a rule that people who joined had to leave the Nazi Party before taking part in any missions, but that didn't mean the Kriegsmarine were passive-aggressively condemning the evils of Nazism. They were mostly just too Christian and conservative to buy into this new-wave Nazi stuff. Still, the Navy was an apt breeding ground for dissenters like Admiral Canaris, who collaborated with generals on the Eastern Front to use a wine bottle to blow Hitler up in 1943.
Which resulted in him being stripped nude and lynched, with his body left up to rot and bloat. Um ... YOLO?
Speaking of the Eastern Front: Invading the Soviet Union didn't go well for the Germans, and the troops' opinion of their supreme commander was directly proportional to the depth of the frozen shit creek they found themselves in. Starting in 1942, the Wehrmacht began taking in Soviet citizens to bolster their ranks, and when Hitler told them to stop recruiting racially inferior people into his army, the officers politely told him to screw off. By the end of 1942, 700,000 of the three million soldiers in the Axis army were Soviets. Additionally, a type of German machine gun called the Sturmgewehre proved invaluable on the Eastern Front ... after it was developed against Hitler's orders.
United States Military
He presumably wanted only pure German weaponry, like schnitzel cannons and bratwurst bombs.
Basically, disobeying Hitler was the main reason the Nazis didn't lose the Eastern Front so quickly that the D-Day troops would have landed in the middle of a Normandy-wide vodka-and-borscht beach party.