Of course not, you'd say. And you'd be right...but. That's the point of this article.
Let's start with the question of what it means for a character to be "smart." I usually think of different kinds of character attributes as contributing to the impression of smartness. The ability to use logic and reason based on experience is one of those. The ability to reflect on and evaluate one's own actions. The ability to communicate effectively can also be interpreted as an indicator of intellect, though it isn't a particularly reliable one.
What the character needs to do intellectually depends on what the story involves. If the character is Sherlock Holmes, he needs to be pretty darned intellectual (along with a host of other things, as watchers of "Sherlock" will testify). If she's out there trying to solve a mystery, she should have the ability to do that, because it will look awfully strange if the protagonist isn't the one to resolve the main conflict. If, as in my stories, she's trying to solve a linguistic mystery, she needs not only mental facility but an expertise in linguistics. There is an endless list of the possible specialties that characters need in order to solve some of those sf puzzles (I've seen chemists, engineers, etc.).
If, by contrast, you're writing something about someone with a divergent mode of thinking - such as the autistic protagonist of The Curious Case of the Dog in the Night-Time - you'll have to gauge carefully how you portray the special characteristics of the person's thinking and then the intellect. They may or may not be related. I struggle sometimes in writing my antagonist Nekantor because he has OCD and is very repetitive - but is also extremely smart, so I have to find a balance in his narrative that makes him plausibly obsessive and plausibly intelligent at the same time.
All right, so let's go a bit further with this. You would put a lot of effort into making sure that you don't make physical or temporal inconsistencies in your story - it's important also to maintain consistency in the intellect of your characters. If a character can use logic to solve a complex problem in one situation, she's very likely to be able to do it in another situation as well. Make sure you don't have your character suddenly lose his mind and behave ridiculously. Sure, there are external influences that can change a character's ability to reason through things (like being in a panic or hurry situation, emotionally distracted, etc.). Just make sure that change is appropriate to the nature of the distractions.
So what do I mean by "does your character have to be as smart as your reader"?
This is a funny one, but to some extent a character needs to share some of the reader's instincts for how stories work.
Particularly if the character is someone who generally comes across as having a strong intellect, or who is having to use reasoning to get through the plot, you don't want (or at least, I don't want) him/her suddenly to catch a case of horror movie stupidity: "Here we are in grave danger from an unknown killing force that seems to come from nowhere but which I've been carefully using evidence to track down - let's split up."
Recently I put down a book because of a different kind of inconsistency. The character I was reading had been spending a lot of time thinking through the motivations of one of his companions, trying to figure out how the guy was thinking, because they were very different types of people - and then suddenly he completely stopped reasoning and decided to take his companion's words at face value (and think worse of him) in a scene where the companion was almost certain to be lying. This is where the reader comes in. Not only was it inconsistent, but I knew the guy had to be lying, and it didn't make sense for the protagonist to think he wasn't. Instead, it gave me the impression (one the author surely didn't intend) that the story was trying to manipulate me emotionally.
In fact, I have run into a very similar situation recently in my own work - where my protagonist has to share some of my own, and my reader's, instinct for where stories usually go. I had my protagonist pursuing someone out toward a waterfall, having some action and argument ensue, and then having the fugitive jump into the waterfall. My critique partner Doug Sharp (yes, he's very sharp) called me on this. He knew the guy was going to jump in the waterfall as soon as he knew that was where they were going, and he was shocked that my protagonist didn't also figure this out.
Oops on me.
So now I'm going to make sure that my protagonist expects that the fugitive will jump into the waterfall, just the way my reader did - and then, I won't have him do it (just to shake things up). After that happened, I caught myself just today realizing I was about to do it again. The previous linguist died mysteriously. Was it an accident? Or was it the evil corporate guys? Well, if you're a reader, it's unlikely to be the former. And if it can't be the latter, then my characters have to entertain the possibility that the natives did it. Because that's the next option for a reader to consider, and not having my protagonist consider it would make him appear stupid.
So while your character doesn't have to be "smart" necessarily, nor should he or she be obliged somehow to match your reader (which would be tough, since you don't know who's going to read it anyway) - it's still good to think through this kind of "knowledge of story/cliché" instinct and make sure your characters aren't dropping the ball at critical moments... or your reader might just drop the book.
And we can't have that!