At this point in Game of Thrones, it's pretty clear that there are just some characters from the books that are never gonna show up on the show - despite their popularity and prominence in the novels by George Rocket Raccoon Martin. The showrunners seem pretty adamant that they're going to finish the show in 7 seasons - and since actually filming everything George RR Martin wrote would probably take 30 seasons, it makes sense that some things are being cut. That's okay though - the show is following its own path, and the more it diverts from the books, the better, because book people will be stuck not knowing what's going to happen for once.
Still, it's a bummer to not have any of these fun, weird characters.
Strong Belwas is a character who was initially introduced alongside Barristan in the books (who remained disguised for a much, much longer period of time). He's mostly just a dude who likes fighting and eating, and has a big ol' belly and a cool-as-hell name.
Could They Be Introduced Later?
There's almost no way to introduce Strong Belwas into Daenerys' story at this point - she already has way, way too many dudes around her to give her advice and/or fight for her (Barristan, Daario 2.0, Grey Worm) that don't really have plots of their own separate from her. It'd be nice to see Strong Belwas as a pal to Dany whose primary motivations for everything are to be able to eat a bunch of liver and onions.
via mathia arkoniel
There are two new major plotlines introduced in A Feast For Crows - the goings-on in Dorne after the death of Oberyn Martell, and the goings-on in the Iron Islands after the death of Balon Greyjoy. Balon Greyjoy is (presumably) still alive on the show at this point, which would mean there's no way to do the Iron Islands stuff, which kicks off at his death. Still, he COULD die very early in the season - but it doesn't seem likely, as season 4 back pedalled on Yara Greyjoy A TON.
We know they're banking hard on Dorne this season (casting many of the Sand Snakes, Doran Martell, and Areo Hotah), which leaves little room for the Iron Islands too. Remember that this show still needs to service its primary cast, which is more fractured than ever. There's Tyrion's plotline in Essos, Cersei and Jaime in King's Landing, Arya in Braavos, Jon and Stannis at the Wall, Sansa and Littlefinger at the Vale, Daenerys and her crew in Meereen, and Ramsay and Reek at the Dreadfort. That's SO MANY PLOTLINES. And they HAVE to keep them all in play, lest they lose the actors (as seen with keeping Theon around to be tortured for an entire season).
No Iron Islands means no Kingsmoot - which means no Victarion being a badass and sailing off to Daenerys with a magic dragon horn. Also, no Euron Crow's Eye being evil and ominous, and no Aeron Damphair being weird and nervous. Theoretically they could show up later, but what would be the point (other than to have a bunch of awesome salty new Greyjoys)? The show's already overstuffed as it is, and there are other ways to get the dragon horn to Daenerys.
via magali-villenueve & henning
This is something weird - a huge portion of the Dorne plot in the books is through the eyes of Arianne Martell, daughter of Doran Martell and heir to the Dornish throne (since women have a lot more rights in Dorne than they do elsewhere in Westeros). She plots an ill-conceived coup with a knight of the Kingsguard and winds up locked away in a tower by her father. Yet even though she's an essential main character, she's not showing up at all this season - all the other Dorne casting has been announced, and younger brother Trystane Martell is now listed as "the heir to Dorne."
Also missing is Quentyn Martell - who really doesn't spend almost any time in Dorne in the books, as he's been tasked with gaining the support and marriage of Daenerys (and her dragons) out in Essos. Given the way his plotline ends though, it's not super surprising that he's being cut, but he was still a pretty major character that's nowhere to be seen here.
Maybe, but it doesn't seem too likely. Arianne's character is probably just being rolled in to one of the Sand Snakes - bastard daughters of Oberyn Martell - in order to strengthen the season's connection to the most popular Dornish character yet (sorry, whoever Oberyn's girlfriend was).
Quentyn seems like he could possibly be introduced - maybe just skip all of his journeying and have him show up at Daenerys' doorstep with the world's lamest marriage proposal.