Ephesians 4:26-27: “Be angry and do not sin; do not let the sun go down on your anger, and give no opportunity to the devil" (cf. Psa. 4:4).

Whether anger is sinful depends on how it is used. It's possible to be angry and not sin. For instance, God is not only angry concerning sin (Prov. 6:16-19), but righteously angry with the wicked every day (Psa. 7:11; cf. Psa. 5:4-5; Prov. 15:9; Rom. 9:13). He legitimately takes righteous judgment upon those that don't repent (Psa. 7:12-14). We call this “righteous anger.” The existence of a hell, a state of eternal separation from God and everlasting punishment, is in itself righteous anger (Matt. 25:41, 46). One of the reasons hell lasts forever is that people don't stop sinning there and therefore God's just and righteous anger continues to burn. [1]

Even Jesus, who was/is sinless (2 Cor. 5:21; Heb. 4:15; 1 Pet. 1:19; 2:22; 1 John 3:5), exhibited righteous anger. Near the time of Passover, Jesus found some Jews treating his Father's temple like a mere house of trade. He took some time to fashion a whip [2]. Only after he completed the whip of cords (an emblem of authority), he drove out the money changers, overturned their tables, and said to them, "Take these things away; do not make my Father's house a house of trade" (John 2:13-17; cf. Mal. 3:1, 3; Matt. 21:12-13).

How we use anger determines whether it is sinful or not. Like Jesus above, legitimate anger towards sin is righteous conduct (cf. Psa. 97:10; Prov. 8:13; Rom. 12:9). However, anger which is rooted within our depravity is sinful. There are numerous cautions against unrighteous anger in the Bible: Don’t be quick to anger (Prov. 14:29; 19:11; Eccl. 7:9; Jas. 1:19). Keep short accounts (1 Cor. 13:5). Don’t allow it to fester over time (Eph. 4:27), rather put it away quickly (Eph. 4:31; cf. Psa. 37:8).

So, anger must be controlled and not used for sinful purposes. Christians should generously forgive others (Eph. 4:32; cf. Matt. 18:21-22).

Reference

[1] The "gnashing of teeth" in Hell (Matt 8:12; 13:42; 22:13; 24:51; 25:30) while possibly referring to pain and suffering, definitely represents anger and seething hatred (Job 16:9; Psa 35:16; 37:12; 112:10; Lam 2:16; Mark 9:18; Acts 7:54-55). Since wicked anger and hatred are sins, then there is perpetual sin in Hell requiring perpetual punishment. (Please see "An Eternal Hell is for Real - The Heresy of Annihilationism?" below).

[2] And when he had made a scourge of small cords,.... That is, Jesus, as the Persic version expresses it. This scourge might be made either of thongs cut out of the hides of beasts slain in sacrifice; or of the cords, with which the owners of the cattle had brought them to this place; or with which they had fastened them in it. And it seems to be made, and used, not so much for force and terror, as to [intimidate], ... the violators of the holy place, deserved the scourge of divine wrath and punishment; as well as to show the miraculous power of Christ in driving such a number of men before him, with so small and insignificant a weapon; for the phrase is diminutive. Gill, John. GiIl’s Commentary: An Exposition of the Old and New Testaments, 6 vols. (Baker Book House, 1980).

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An Eternal Hell is for Real - The Heresy of Annihilationism?