Burnet explains that μὴ οὐχὶ as dependent on μαλθακοῦ εἶναι ἀνδρός, which “implies a negative and therefore takes μή οὐ.”
Radically simplifying in the interest of showing the base construction: ἐμοὶ δοκεῖ < τὸ μὴ οὐχὶ ἐλέγχειν > μαλθακοῦ εἶναι ἀνδρός. It appears to me that not refuting is an act of a cowardly man.
Since εἶναι is not negated, the μή οὐ is surprising, and would be what Burnet is trying to explain by “implies a negative”.
The relevant discussion from Smyth would appear to be 2748 under the heading "μὴ οὐ with the Infinitive depending on Negatived Verbs,” and discussing an exception to the normal rule:
Smyth mentions explicitly only πᾶσιν αἰσχύνην εἶναι, αἰσχρόν ἐστι, and δεινόν ἐστι, but μαλθακοῦ εἶναι ἀνδρός is obviously the same as the others. See 2748d for the τὸ μὴ οὐ variation from simple μή οὐ.2748. Some expressions denoting repugnance to the moral sense involve a negative idea, and may have the same construction. Thus, ὥστε πᾶσιν αἰσχύνην εἶναι μὴ οὐ συσπουδάζειν so that all were ashamed not (i.e. felt it was not right) to coöperate zealously X. A. 2.3.11. So with αἰσχρόν ἐστι ( = οὐ καλόν ἐστι), δεινόν ἐστι
What, however, is the negative implied in these statements that makes μαλθακοῦ εἶναι ἀνδρός and so on equivalent to a negated verb? Burnet doesn’t say, and Smyth writes “= οὐ καλόν ἐστι” as if that explains something. It does not. We can write down the opposite of any statement, but that doesn’t tell us why these get μή οὐ but others do not.
My guess is that the implied negative concept is actually something like “and you are not a coward” or “and they were not the sort to do shameful things”, both of these examples (Phaedo 85c, and X. An. 2.3.11) being exhortations to the opposite action, not simple statements of actual shame felt from a completed action. But I am away from my library this week and don't have access to any full list of examples involving Smyth 2748 equivalents.