Question: Recently I walked through Dolomites in Italy, and there was a warning sign that this is bear-encounter area. There were tips on what to do in general. But one of tips was "if bear approaches you aggressively, don't move and be passive".

Bears in this area are brown.

Why? If bear wants to eat me or kill me, I would prefer to make it at least a bit more difficult to him. I know I will die anyway, so why wouldn't I at least try to die "like a manly man (TM)".

Answer: 
Brown bear attacks tend to be defensive attacks, where the bear is trying to protect its cubs, territory or food. The bear usually isn't attacking you because it wants to eat you, but is trying to neutralize a perceived threat. By playing dead, a brown bear may not perceive you as a threat anymore, and may leave you alone. Fighting back may make the bear more aggressive, as it senses a greater need to defend against a dangerous creature in its territory.

Black bear attacks, on the other hand, tend to be predatory in nature - the bear is attacking you with the purpose of killing and eating you. Playing dead in this scenario won't help, it'll just make you an easier lunch.

The common saying about bears is, "if it's brown lie down, if it's black fight back, and if it's white say goodnight". Just hope you're not attacked by a polar bear.
When faced with a brown bear ( in Wyoming, near Yellowstone ), while on hands and knees in a small tent ( middle of night) ; I gently but firmly said "shoo bear". I thought a loud noise might upset it. After repeating shoo several times he walked away. It was good that he had already taken anything he wanted out of the cooler chest he broke open. ( They do not like tomatoes or mustard). This may not have been the recommended action , but the bear was 10 ft away and the car was about 30 ft, I thought running was a low odds choice.