In the burton series of batman, the connection between all the films is rather minor...other than having the same character returning to the film. What happens in the previous film has very little impact in the next film.
In the dark knight, it is the first batman sequel that really explore the consequences of the previous film. What happens in BB result in what has happened in the next film.
In fact at the beginning of BR, Bruce Wayne is very frustrated and Batman is quite cynical and distrustful, and that’s because the motive of his mission has vanished; the kliller of his parents has died and his revenge is accomplished. Sure, he stills goes when there are troubles but it’s not quite the same as before.
The difference with TDK is that in BR they didn’t explain all this in detail.
Now after BR it was all a new thing. Bruce Wayne – who had had this big burning love with Catwoman – suddenly feels attracted to a teenager-minded chic called Chase. “I have never been in loive before Alfred.” Can’t buy that.
For example, the amount of crazies working for the joker, the need for batman in gotham, why he can't take off his mask, why bruce wayne isn't living in his mansion and etc.
All of that is cool. But in all seriousness, how many people actually need an explanation as to why a villiain in a Batman story has henchmen (for decades this is a given; villiains do have henchmen), the need for Batman in Gotham (he’s the hero fighting crime, which all cities have), why he can't take off his mask (seriously you don’t need a whole movie to figure out why superheroes’ secret identity is important to them), why Bruce Wayne isn't living in his mansion (this could be intriguing, but where Bruce lives doesn’t affect the story much. It’s more like nothing actually).
So, do you think that TDK requires you to watch BB first to really enjoy this film?
It’s always better to watch the whole saga, but it may not be mandatory in this case.
The only reason why you should be aware of what happens in BB is Rachel. The relationship between her and Bruce is the only thing that might be necessary to be known.