Well, that's a bad way to phrase the question: Do characters have to change? No. Thanks for coming, see you next week, and tip your waitress.
Maybe ask it this way: Do readers/viewers respond more when the character faces some crisis and changes? That's a little more complex a question and I'd like to look at two action/adventure films that I genuinely like and see what role character growth plays in my enjoyment. It helps that both films feature big guys punching Nazis.
Hellboy is an interesting film in that it flirts for a while with multiple protagonists: Hellboy seemingly has to face up to his freakish nature and human choices, while Agent Myers seemingly has to deal with the strangeness of the world. I say "seemingly" because both of these plot lines are really quickly handled or not really addressed. Hellboy starts off as a big lug whose solution to things is to hit them very hard; he ends up as the same. Myers may start off as the naive viewpoint character, but after one "what is that thing?" and one "I don't belong here," we never hear a peep about that.
Even Hellboy's major crisis moment is pretty weak sauce; when the movie presents the question "will he or won't he help Rasputin call down the Ogdru-Jahad?," not only is the answer an obvious "NO," but that answer remains pretty solid throughout the movie. There's one brief moment where Hellboy helps Rasputin and we see some cool special effects about what that means--and good thing too, because otherwise we might wonder why Hellboy is suddenly helping the guy who killed his dad and his love interest.
In fact, if we wanted characters who seem to change, we'd have to look to the director of the Bureau (played by Jeffrey Tambor) and to Liz Sherman. The director starts out as a bureaucrat/spokesperson; he ends as a field agent. He begins with a distrust of Hellboy as just another freak; and he ends working with Hellboy to kill a supernatural Nazi and then teaching him to light a cigar. Can we say here that a cigar isn't just a cigar? It's clear that Tambor's character grows from disapproving to supportive father figure. Why he changes is a little less clear to me; and after that support, he's left behind, another reminder of his symbolic role. (Another reminder: I don't even remember his name.)
Liz Sherman also changes pretty quickly, from person who doesn't trust her powers and thinks it would be weird to date Hellboy; to person who uses her out-of-control powers and kisses Hellboy. That first change takes a while and is fairly played out, with Myers (and Abe Sapien) building up a trusting relationship. The second is a little quicker and stranger, seeming to be based on the "you'd do that for me" moment of Hellboy killing monsters for Liz. (Of course, he'd do that anyway.)
So while Hellboy doesn't really change all that much, the movie presents an illusion of character growth by having the characters around him consider him differently.
I did say I was going to look at two movies about big guys hitting Nazis and whether the main characters actually grow at all. With Captain America, the case seems easier: (a) there's no one else to compete for screen time with him the way Myers competes with Hellboy; (b) he undergoes some serious physical change.
But that physical change that makes Steve Rogers into Captain America only brings out the heroic qualities that were in Steve to begin with. And the movie does this brilliantly, setting up his inner heroism in a bunch of smaller scenes that make his later heroism believable as part of his character. In other words, at the end of the movie, Rogers chooses to sacrifice himself to save New York; and we know he would do the exact same thing at the beginning since we've seen it in much smaller ways--like when he sacrificed himself to fight a bully so that other people could enjoy the movie and honor the troops in the newsreel. In other words, there's no internal conflict, no crisis of character for Captain America to face. He starts off as a boy scout and he ends up as a boy scout.
There is one way that he does change, though: as the movie goes on, he gets better at talking to girls. Specifically, he gets better at talking to Peggy Carter. When we first see Rogers with girls (at the World's Fair), he has no interest in women; later, Carter even calls him out on his awkwardness when she takes him to the experiment; but by the end of the movie, after they have the requisite "this isn't what it looks like" misunderstanding, he's flirting right back with her. That's something he couldn't have done before.
And again, while the core of Steve Rogers doesn't change, we get some illusion of change through the fact that other people look at him differently. In fact, it's the Colonel (played by Tommy Lee Jones) and Peggy Carter that seem to look at him differently--and they occupy the same positions as the characters in Hellboy: the boss and the love interest.
The villains' story
Here's another interesting parallel between Hellboy and Captain America: both start with a tiny prologue introducing the main character--but not really: we see Hellboy as a baby and Captain America as a popsicle. (Actually, we don't see him frozen, but we see his shield). And then both movies introduce us to the villains, whether Rasputin reborn or Red Skull finding his cube.
This early focus on the villains might threaten to make them the center of our attention, except (a) they're both marked as EVIL; and (b) despite some cool character design, these guys never really become interesting. Rasputin wants to summon dark gods and Red Skull wants to control the world. Why? No reason--or backstory--is given. What interesting positive traits do they have? Well, Rasputin loves his she-wolf, Ilsa; and Red Skull is clever enough to find the cube. But there's nothing really surprising--no, "Rasputin loves puppies" or anything like that.
Character change is often considered a requirement of good story-telling; but at the same time, in action/adventure movies, we often like our protagonists to start off as good guys. So the change in these two movies isn't in the essential goodness of the characters, but in some other quality--or sometimes just in the illusion that change is taking place by having supporting characters change their opinions.