Thoughts on what Bioware can take away from FO4?
The value of trying to experiment and do old things in a new way. I applaud that sort of innovation- and Bethesda trying to improve its dialogue and inventory system is exactly the sort of effort I hope Bioware keeps applying.
What, specifically, to emulate? Not too much, actually. Hair in the character creator, I guess, but FO4 is more about what Bethesda is trying to learn and take away from Bioware. Character companions, dialogue improvements, and plot framing. Still, the world vs. narrative emphasis is strong, and not great for a Bioware plot model.
Avoiding things that don't work well in a Bioware game would be the Bioware lesson- which includes the inventory system of FO. Stay away, please. Meanwhile, Bethesda can stand more practice on its elements.
"Sarcastic" 
That probably sums up the best of the worst of the dailogue system- and the good example of how a voiced protagonist can be done wrong. It doesn't help that you can awkwardly walk into and out of conversations.
That said, while I miss the voiceless protagonist, I will admit it has charm in helping encourage an emotional conversation rather than the one-way exposition characters. Between FO3's 'Find the Father' narrative and FO4's 'Find the Son', protagonist investment is more credible in the second. And the early part of the game- the pre-war sequence of the married household- wouldn't work as well with a silent protagonist.
On the other hand, I miss the factional reputation system. I'm still trying to get a handle on the companion opinion systems, and while I can tell there are factions (and I assume some will be mutually exclusive), it's hard to track where I stand with them.
The game is quite literally asking you to collect garbage. This feature is the essence of, IMO, what's wrong with crafting. I can appreciate that some people enjoy this time of game. But there had to be a way to balance between a feature like that and literally walking through town collecting and recycling garbage.
I feel it's an improvement on the Bethesda typical model for items.
Bethesda games have always been a trash collection game- scavenging whatever has value to bring back and sell for caps. Between limited inventory (weight) and limited caps with vendors, it's always been a balance of weight-value. Thing was, most junk in Bethesda games was simply that- total junk. And there was so much of it, picking it up clogged your inventory right away, and it almost never served a purpose. As a rule, I never picked up anything in Fallout 3/NV that wasn't 20 caps/lb.
Games like FNV and Skyrim tried to make up for this lack of value with their crafting systems. Then, even low-value trash items could become valuable and have use. Turning ore into ingots into armor. Or taking a hide and tanning it for higher value. Then there was a point for collecting those lower-value items- but the trick was, you had to know which ones you were looking for. By and large, they weren't interchangeable.
FO4 still has item-specific recepies, but the changes to the junk system gives all junk at least some utility value by virtue of breaking down into the resource components. There's no longer 'trash except for crafting' and 'utter trash'. And the system has been jiggered to help support it- the trash item category makes it easier to consolidate and unload all that trash, rather than having trash items interspaced with all your good items like they were in previous games. Once you got your settlement up and running, going to your settlement is the convenient place to unload your trash (one button) and then sell the valuable stuff.
But, above that, I feel the item system for is better for supporting a theme of the more recent games- of building a post-apocalyptic future and rebuilding civilization. Crafting is a great mechanic to support that theme, and the use of junk of the Old World into new construction is thematically appropriate. Previous fallout games were more limited in that- FO3's use of junk was for the sub-par rock-it launcher- but even Vegas had limits.
The settlement creation system is a great step forward in that overall Fallout theme. It needs a lot more- personnel management in a settlement, rotation of items for placement, inter-settlement interface- but it's a great step forward.