BE has its issues, but it has a lot of good points as well. To return to the intent of the original post, I think it there are some things that it is particularly useful to learn from it.
Problem: The end game is pretty awful. Specifically, I build the Mind Flower or whatever the Harmony end-game building is which take twenty-something turns, and then I am told it will be 39 turns until I win. The AI does not proceed to get any more aggressive, and my military is too dominant for it to matter anyway. This means I have won already, but the game is making me hit end turn 39 more times before the game actually ends. This was probably the most boring sequence I've encountered in a 4x game. Oh, and there isn't even a cut scene at the end.
Lesson: All win conditions should be fun and the once victory is assured, the game should end quickly. Any win condition that ends with hitting the turn button over and over again should be redesigned. Also, I know it seems silly, but a cinematic for a win goes a long way toward creating a sense of accomplishment at the end.
Problem: Bad AI. I am not a particularly skilled 4x player, and am usually not too critical of AI, but the BE AI does some glaringly stupid things. I would go to war with a civ across the world and we would never get our troops close to each other's territory, then he would send me a peace proposal offering once of his cities.
Lesson: Not too worried about this one for GC3, if GC2 had any specific strength relative to other 4xs it was its AI, and I trust that GC3 will be the same.
Problem: Health. There may be some who disagree with me, but health in BE and its Civ 5 equivalent happiness drive me up a wall. In theory, this is a mechanic designed to slow/stop city spam and steamrolling. In practice, what it does is punish the player for active, aggressive game-play. Essentially you are punished for fun.
Lesson: Don't punish active game-play. There is nothing in GC3 so far that resembles this mechanic, but the lesson should be kept in mind more broadly. Think about how your mechanics encourage players to play, and make sure that that play-style is fun and engaging.
I think it is also useful to look at what BE does well.
Positive: I think the thing that the greatest strength of the game is how its various systems interact. Tech, ideology, and virtues all interact in meaningful and interesting ways. The tech web is interesting, and I think more engaging than a traditional tree, although it can also be very confusing at times. I really like the way virtues work, with the x and y axis bonuses. I also like how tech gives you different kinds of units, while ideology gives you stronger units. That kind of interdependence of systems is strategy gold. Ideology also interacts with resources in a way that is interesting, but I don't always like. It's cool that your over-all strategy is that tied to the terrain, but it can be frustrating as well. I want to play a supremacy game, but in 5 games have never started anywhere close to Firaxite and its a little frustrating to be making the big ideological decisions based on the luck of map generation.
Lesson: The biggest take-away for GC3 specifically is for their ideology I think. The more integrated it can be with the other game systems the better. It should change the way you play, not just be a list of bonuses.