What do you do with a comma that would come after a dash?

In short, I would either leave out the comma before but or, if that bothers you, simply use different punctuation—such as parentheses.

Let's start from the simpler sentence and then add the extra information:

I am a man, but I cannot endorse your claim.

So far, so good.

Now, we want to add a strong, burly man as parenthetical information:

I am a man, a strong, burly man, but I cannot endorse your claim.

While this is syntactically sound, the proliferation of commas makes it look a little odd. So, in order to deal with all of the commas, it can be rephrased in a few different ways:

I am a man, a strong and burly man, but I cannot endorse your claim.
I am a man (a strong, burly man), but I cannot endorse your claim.
I am a man—a strong, burly man—but I cannot endorse your claim.

The problem posed in the question is whether or not there needs to be a comma placed before but—something which would typically (although not always) be the case when there are two independent clauses.

Let's take the example with the parentheses:

I am a man (a strong, burly man), but I cannot endorse your claim.
→ I am a man, but I cannot endorse your claim.

Here, we remove all parenthetical information including its punctuation, and end up back at the original sentence.

Now, let's look at the version that uses parenthetical commas:

I am a man, a strong and burly man, but I cannot endorse your claim.

This version also seems fine in long form. Yet, if we were to actually remove the optional information and its punctuation, we would end up with this:

I am a man, a strong and burly man, but I cannot endorse your claim.
→ I am a man but I cannot endorse your claim.

Note that there is no comma—because it was removed. But I suspect nobody would think the long version is incorrect because of this.

When the extra information is taken out, we simply don't remove both commas. Depending on how you look at it, we either keep one of them or we remove both—and then add one back in again:

I am a man, a strong and burly man, but I cannot endorse your claim.
I am a man, a strong and burly man, but I cannot endorse your claim.
I am a man, a strong and burly man, but I cannot endorse your claim.

→ I am a man, but I cannot endorse your claim.

In fact, you could make the claim that the second comma in the full sentence is playing a dual role. It's serving not only to terminate the parenthetical information but also to introduce the second independent clause.

This follows the same rule as when we end a sentence with an initialism:

✘ I live in the U.S.A..
✔ I live in the U.S.A.

Rather than repeat the period after A, we only use a single period. It serves a role in both the initialism and in the termination of the sentence.

The use of the comma has the same kind of dual role:

✘ I am a man, a strong and burly man,, but I cannot endorse your claim.
✔ I am a man, a strong and burly man, but I cannot endorse your claim.

We do not use two commas in row, but only a single comma.

With that in mind, let's look at the version of the sentence that uses dashes:

I am a man—a strong, burly man—but I cannot endorse your claim.

I would argue that the second dash is, as with the single period after the initialism and the single comma after the parenthetical commas, serving a dual role.

It's not only acting to terminate the parenthetical information, but it's also standing in for the comma that would normally go before the conjunction.

Therefore:

I live in the U.S.A [..]
→ ✔ I live in the U.S.A.

I am a man, a strong and burly man [,,] but I cannot endorse your claim.
→ ✔ I am a man, a strong and burly man, but I cannot endorse your claim.

I am a man—a strong, burly man [—,] but I cannot endorse your claim.
→ ✔ I am a man—a strong, burly man—but I cannot endorse your claim.

However, if you don't like how that looks, then simply use parentheses for situations like this.

A final alternative is to not make the extra information parenthetical at all:

I am a man—a strong and burly man, but I cannot endorse your claim.

Here, I've again replaced the comma between strong and burly with and (to avoid the appearance of too many commas). But the entire phrase a strong and burly man is no longer parenthetical; instead, it's actually additional.

This changes the meaning of the sentence very slightly (it's no longer optional information that can simply be removed), but that may be acceptable.