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I say that the "that" in the original question is flatly wrong. It is a foreignism. I'm not an expert on why, but I will try to find out.
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He eats whatever is sweet.

He eats whatever that is sweet.

In the example above, whatever is classified as a compound relative pronoun. The compound relative pronouns function simultaneously as the relative pronoun and its antecedent. Other examples of compound relative pronouns are what, whoever, and the archaic whosoever, whatsoever, whoso. These archaic ones might remain familiar through their use in the King James Version of the Bible.

In all of the examples above the that is a relative pronoun.

When using these compound relative pronouns, you cannot add another relative pronoun such as that. There is no need for it. It's wrong.

Say what you think. (and not: *Say what that you think.)
I like whatever you write. (and not: *I like whatever that you write.)
She fires whoever disagrees with her. (and not: *She fires whoever that disagrees with her.)

In these uses as relative pronouns you should be able to replace them as follows:
what = "the thing that"
whatever = "anything that"
whoever = "anybody who"

Not to muddy the water, but you have to be careful to distinguish between that as a relative pronoun and that as a demonstrative pronoun or adjective as in the following correct sentences:
I asked her to document whatever that does.
He obeys whatever that woman says.

In the above sentences, the that is functioning as a demonstrative pronoun or adjective, and not a relative pronoun.

Bravo, Glenn.   I never learned about compound relative pronouns in high school English, but you explained it perfectly.   I am now wondering about Bob's list.   Going strictly by how it sounds to my ear - everything in the list sounds okay to me except the last two (adverbial prepositions?) Can anyone tell me how "that" is used in these examples. Bob talks about what it does but he doesn't define it's use.

I second Dick's bravo, Glenn; very informative, very convincing.   Now (if you don't mind) I want to explore relative pronouns a bit more; in the course of reading Glenn's post and looking up relative pronouns, I became convinced that my examples above don't fit the definition.

I suppose relative pronouns are so called because they relate their object to something elsewhere in the same sentence.   I was unable to find complete consensus on the terminology, so I'll invent one for this discussion (but feel free to suggest your own terms):   In the following example the relative pronoun is in italics, its object is underlined and its antecedent is bold.

The Camry that ended up in the ditch was stolen.
The Camry whose seat was torn was stolen.
The Camry that he stole ended up in a ditch.

Will the person who owns a silver Camry please come to the front desk?
The person whose Camry was stolen ended up in a ditch.
The person whom I threw in the ditch ended up with his seat torn.

Glenn described relative pronouns accurately, and in reading his post and looking up what still bothered me I learned some things I either never quite locked down before or forgot after learning.   But none of the examples in my earlier post are relative pronouns—or at least they have no antecedent that I can discern, and "that" seems to fulfill a different function in all of them.   It's true that one of the sites I looked up gave this sentence as an example of that as a relative pronoun:

I cannot believe that he said it.

But it's wrong.   Glenn, or someone, please take a closer look at this example and at mine above and see whether you can find out what this usage of "that" should be called; I've no idea.

Bob, here's one more that:

      He climbs rock with no ropes- he's that good.

This that sounds like same as so.  

I agree about "that" being an adverb, in that position; I suppose it's a minor modification of its function as a demonstrative adjective.   In the sentence "That dog is brown", "that" is an adjective; in "he is that good" it's an adverb, ie it modifies an adjective.   (I'd feel better if we had different terms for verb modifiers and adjective modifiers, but ok.)

As for it being a conjunction in "I believe that I'll have another drink", I didn't like it at first but maybe I should give it a chance.   My reflexive complaint is that it doesn't join but subordinates "I'll have another drink" to "I believe".   But some conjunctions do that, such as "although".

Still, something different is going on here:   "that I'll have another drink" is the object of "believe".   Seems to me that isn't the usual function of a conjunction.   As I said, it wraps up a verb phrase as a package, a substantive, that can then serve as the object or subject of another verb.   What do we call that?

It also has that function, come to think of it, when it packages a verb phrase as an indirect object or as the object of a preposition:

I wanted to alert you to the fact that I'll be out of town next week.

We'll meet to talk about the fact that you haven't been doing your homework.

After a preposition we use not "that" by itself but some longer construction like "the fact that" or "the proposition that" or "the theory that".   But its purpose is the same in each case:   It bundles up the following verb clause as the object of that preposition.   If that's a conjunction, it's not like any other conjunction I can think of.

The object of a verb in each of your examples is a noun clause, sure, but "that" doesn't appear in any of them.   In my examples, "that" packages a verb clause up as a noun clause.   Now, conjunctions conjoin verb clauses:

He went there and he did that.

I would like that but I'll have to check with my wife.

It's a good idea, although I'm not sure it's the best one.

Some conjunctions, such as "although" above, do indeed subordinate one verb clause to another.   But "that" turns a verb clause into a noun clause and makes it the object of a verb, which doesn't strike me as the same thing.

I don't have an term for it that I trust; but I'm unconvinced by anything else I've heard so far proposed, too.

Ok, "subordinating conjunction", then.   But in that case maybe I had the wrong idea of what a subordinating conjunction is.   I was thinking of conjunctions such as "although", "if", "because" and the like; maybe I was wrong.   This, at any rate, is different.   Are there other subordinating conjunctions like it?