#1. The player cannot anticipate story consequences of their class when they choose it at the beginning of the game.
#2. Different classes leading to even minor different story outcomes (such as different minor characters surviving or dying) compels players to play classes they don't enjoy to get the story they want.
#3. The player's class affecting the story is almost certainly going to necessitate the player's companions classes' affecting the story. This leads to a whole subset of fresh problems as to the player being punished or rewarded for being the 'right' or 'wrong' companions, as well as the composition of the cast.
#4. Characterization which would otherwise be the player's choice or available to all classes would likely be restricted to certain classes. Players are again compelled to play a class they may not like to get the personality they actually want. It addition, players are likely to find the 'best' personality as a mix between classes and thus off limits.
#5. The active use of class in cutscenes is almost guaranteed to create enormous problems. Again, this can be divided into whole subsets of problems on their own, all of which need to be addressed. The central problem is how can developers possibly script legitimate combat cutscenes when a huge variety of abilities are available to the player character? Other problems include the expectation of class abilities, which I can almost guarantee is going to lead to a decrease, not increase of 'immersion' and a surge in complaints over non-existent 'plot holes,' and the simple problem of thinking up legitimate animations for the more obscure classes such as sentinels.
Yeah. Very, very, very big problems.