"too dumb not to ~"

Solution 1:

The sentence is an example of saying one thing and meaning the opposite thing.
It is called antiphrasis.

So, "too dumb not to run away" means: "not smart enough to run away".

It really is as simple as that. This usage is very common in everyday speech.

antiphrasis

Solution 2:

Context drives how the negation is interpreted.

I usually don't link to Tumblr posts as evidence, but this post by user snowgall offers a well-sourced explanation of how the statement means that Steve Rogers doesn't run away from a fight. The hypothetical examples are knotty, but the actual examples help illustrate its use:

Data is always a good thing to keep it real, so here is a parallel example from a work in print:

• “Soon would come dusk-calls of birds too dumb not to leave a war zone. Or too weary of searching for places in Nam without war. He and the other four were missing chow now. But Honda or Philly would have got food from home, and besides …” Prism International 1991 - Volume 30 - Page 73

It’s clear from context that the intended meaning is that the birds should leave the war zone, but they are not doing so because they are too dumb.

This is just like: Steve should be running from a fight, but he’s too dumb to do that, so he fights.

Here’s another example:

• “Our parents wouldn’t let us get married in high school, and we were too young and too dumb not to know that that was excellent advice” Kansas Quarterly 1987 - Volume 19 - Page 12

So there seem to be two options for interpreting the statement.

1. Read "not" as a logical inverter, as the question-asker does - "too dumb not to run away" could be logically reduced to "too dumb to fight" ("not" applied to "to run away").

2. Read "not" as an intensifier where "dumb" and "not" both already have negative senses: "too dumb not to run away" is an intensified version of "too dumb to run away."

If Bucky or the filmmakers were trying to write technically precise English in the absence of further context, they would have failed. However, in the context where Bucky is responding to Captain America asking him if he was "ready to follow Captain America into the jaws of death," where both speakers and the audience know Steve Rogers doesn't back down from a fight, the second reading is much more likely.

Solution 3:

A phrase like "he's too ADJECTIVE to VERB" means that the person does not VERB because they are ADJECTIVE.

For example:

She is too smart to care -> She does not care, because she is smart
He is too short to reach the shelf -> He cannot reach the shelf because he is short

The structure is more complicated (and thus more rarely used) with negative statements, but they're still possible:

She is too smart not to notice -> she does not not notice (she does notice) because she is smart
He is too short not to fit in the car -> he does not not fit (he fits) because he is short

So, following this same pattern:

He is too dumb not to run away -> he does not not run (he runs away) because he is dumb.

That's a strange idea (smart people do run away from fights) and apparently contradicts what we know about his character, which makes it seem like a mistake. It's grammatical, but it's probably not what the speaker really wanted to say.