Eph 4:26 Be ye angry and sin not: let not the sun go down upon your wrath.
Prov 22:24 ...make no friendship with an angry man, and with a furious man thou shalt not go.
In order for this to be a true problem, it must be established that anger of any form is sinful and wrong, or, it must be established that the type of anger in Prov 22:24 is the same type of anger in Eph 4:26. Can a strong case be made for either of these possibilities? More is needed than the implied assertion that both verses in question contain the words ["angry" and "anger"] which have the exact same root concept.
The reader should note that the crux of Eph 4:26 is the Greek orgizesthe is translated by the KJV as a mere imperative: "Be ye angry". Yet, the NIV takes orgizesthe as a more permissive imperative "Do not let the sun go down while you are still angry...".
The Blass-Debrunner-Funk grammar, section 387, views this as a concessional imperative: "You may be angry as far as I am concerned (if you can't help it), but do not sin thereby." Robertson's A Grammar of New Testament Greek in the Light of Historical Research , page 949 agrees with the BDB grammar. See also Robertson's Word Pictures in the New Testament , volume IV, pages 540-1.
If Robertson and BDB are correct there is absolutely no case for a contradiction here. However, Wallace's Greek Grammar Beyond the Basics , pages 491-492 (see footnotes 110-3 as well) does not seem convinced here and seems to conclude that the strongest view is that this text "seems to be a shorthand expression for church discipline, suggesting that there is biblical warrant for dikaia orge (as the Greeks put it) -- righteous indignation."
The commentaries as well as the divided between the view of Robertson/BDB and that of Wallace. So the exact sense of what Paul says here can mean one of two things: a concessional imperative, or a strict imperative. If the concessional imperative view is adopted, then the case for contradiction is dogmatically incorrect.
But suffice it to say that at the very best for the skeptical charge there is no convincing evidence at all for a contradiction between these two passages, just the mere claim.