Have There Been Any Important Characters Without a Strong Heart?

I'd agree that it isn't since Beast's heart is strong because of his love for Belle.

 

Ahh! I see thanks a lot!

 

none come to mind, no.

but if you ask me we really do need some, for variety.

When all of your main characters have notably strong hearts it, for one thing, makes the protagonists (Sora, Riku, etc) seem far less significant than they should be, and also limits what you can do with the characters who shouldn't have strong hearts. It would be far more interesting to have a few characters with strong hearts and some with weak hearts so that the ones with weak hearts can get their own character moments and become worthwhile characters in their own right, something like you don't have to be strong to be important.
I still think this should have been how they wrote Kairi, but that's obviously not how they did.

This pretty much all goes back to how I don't like how almost everyone has a keyblade.

 

I totally agree with the notion of "you don't need a strong heart to be powerful," I would love to see more of that in Kingdom Hearts. I guess, though, you could make an argument for Ven--his heart, though pure light, was seriously fractured by the production of Vanitas. That's why Eraqus (and, by extension, Aqua) worked so hard to protect him.

However, while a character may have a strong heart, that doesn't mean they're not susceptible to strife. I was originally going to post about Riku and Cloud--both of them have immense struggles with the darkness; Riku in particular has proved immensely susceptible to reckless curiosity and jealousy. However, upon consideration, their ability to overcome these obstacles is what proves their heart strong, in my opinion. Even consumed by Ansem SoD (after Sora stabs himself), Riku is able to hold off SoD to protect his friends; in Chain of Memories, he is able to find the strength to not give up and to walk the path of twilight.

Riku may be unique in that case, though. Some of the other characters--Sora and Kairi in particular--seem to have rather shallow characterization in comparison; we are told they have strong, pure (relatively in Sora's case) hearts rather than shown the strength of their hearts, as we were with Riku. In a story it's important to have a "pure" character, I think, and Sora can act as a terrific foil to Riku. Kairi follows the "Princess" archetype, Sora the very Square-Enix valued "happy-go-lucky, positive if a bit daft, hero" (Tidus in FFX being another example of this). But it is also a great storytelling technique to deconstruct these types of characters--I don't predict we will see it much with Kairi, because (following Kingdom Hearts lore), her heart is pure light, though she may develop into more of a warrior role. Sora, on the other hand, I think has great potential here, particularly with the ending to Dream Drop Distance.

 

Sora's ability to have a keyblade could have sufficed as a sign of his heart's strength, but after BBS that's no longer valid.
Like I've been saying, now that that's invalid and everyone else has a keyblade (having to type this phrase over and over is making my physically sick) there is nothing to distinguish Sora as someone significant, but, like you said, after the end of DDD there is a big opportunity for them to develop the character.