It might be noted that the adverb εἰκῇ (eikē) is not terribly common. It only appears in 6 other places in the New Testament, where in all these it is translated in the KJV as in vain (or vainly, Col 2:18); and only once in the entire Septuagint, where Brenton translates the word as rashly:
Proverbs 8:25 LXX
ἄπληστος ἀνὴρ κρίνει εἰκῇ, ὃς δὲ πέποιθεν ἐπὶ κύριον, ἐν ἐπιμελείᾳ ἔσται
An unbelieving man judges rashly: but he that trusts in the Lord will act carefully
It also appears in the writings of Clement of Rome (b. 35), who in his first Epistle (40:2) writes:
τάς τε προσφορὰς καὶ λειτουργίας ἐπιμελῶς* ἐπιτελεῖσθαι καὶ οὐκ εἰκῆ ἢ ἀτάκτως ἐκέλευσεν γίνεσθαι, ἀλλʼ ὡρισμένοις καιροῖς καὶ ὥραις
Now the offerings and ministrations He commanded to be performed with care, and not to be done rashly or in disorder, but at fixed times and seasons
where in the translation above Lightfoot also chooses the word "rashly"
Thus, εἰκῆ might be better translated "without good cause" rather than simply "without cause".
As previously noted, Metzger's Textual Commentary on the New Testament maintains that εἰκῆ was added by copyists sometime after the 2nd century "to soften the rigor of the precept".
That having been said, however, it appears that several Ante-Nicene Church Fathers quoted the verse with εἰκῆ included. These include Irenaeus of Lyon (b. 130; Against Heresies, IV.13.1, IV.16,5) and Cyprian of Carthage (210-258: To Quirinius, III.8). Neither of these early Church Fathers were particularly noted for their lack of rigor. The wording is also found in the Diatessaron of Tatian* (VIII.51), a Syriac harmony of the Gospels which dates to late 2nd century, as well as the Apostolic Constitutions (LIII), which have an indeterminate but ancient date. The verse is quoted without εἰκῆ by Origen (184-253; not generally considered a Church Father) in On the First Principles (III.1.6, IV.1.19).
In his book, Of the Spirit of Anger, the later Church Father John Cassian (360-435) directly addresses the "controversy" of whether without a cause should be included, in Chapter XXI ("Whether we ought to admit the addition of without a cause in that which is written in the Gospel, whosoever is angry with his brother"). He writes:
But you should know that in this, which is found in many copies, Whosoever is angry with his brother without a cause, is in danger of the judgment, the words without a cause are superfluous, and were added by those who did not think that anger for just causes was to be banished: since certainly nobody, however unreasonably he is disturbed, would say that he was angry without a cause. Wherefore it appears to have been added by those who did not understand the drift of Scripture, which intended altogether to banish the incentive to anger, and to reserve no occasion whatever for indignation; lest while we were commanded to be angry with a cause, an opportunity for being angry without a cause might occur to us. For the end and aim of patience consists, not in being angry with a good reason, but in not being angry at all. Although I know that by some this very expression, without a cause, is taken to mean that he is angry without a cause who when he is angered is not allowed to seek for vengeance. But it is better so to take it as we find it written in many modern copies and all the ancient ones.
On the other hand, a near contemporary of John Cassian's, John Chrysostom (d. 407), makes a case for including the phrase. Commenting on Paul's 2nd Epistle to the Corinthians, he makes the argument that anger is useful (and presumably justified) "for the succor of the innocent":
Therefore also Paul said, It is better to marry than to burn (1 Cor. 7:9) and Christ, He that is able to receive it, let him receive it (Matt. 14:12). But concerning money He spake not so, but, whoso hath forsaken his goods shall receive an hundred-fold (ib. 29). "How then," saith one, "did He say of the rich, that they shall hardly obtain the kingdom?" Again implying their weakness of character; not the imperiousness of money, but their utter slavery. And this is evident also from the advice which Paul gave. For from that lust he leads men quite away, saying. But they that desire to be rich fall into temptation (1 Tim. 6:9), but in the case of the other not so; but having separated them for a season only, and that by consent, he advises to come together again (1 Cor. 7:5). For he feared the billows of lust lest they should occasion a grievous shipwreck. This passion is even more vehement than anger. For it is not possible to feel anger when there is nothing provoking it, but a man cannot help desiring even when the face which moveth to it is not seen. Therefore this passion indeed He did not cut off altogether, but added the words, without a cause (Matt. 5:22). Nor again did He abolish all desire, but only that which is unlawful, for he saith, Nevertheless, because of desires, let every man have his own wife (1 Cor. 7:2). But to lay up treasure He allowed not, either with cause or without. For those passions were implanted in our nature for a necessary end; desire, for the procreation of children, and anger, for the succor of the injured, but desire of money not so.
Theophylact's later commentary on the passage emphasizes the difference between what we would describe in English "without cause" and "without good cause":
He who is angry with his brother without good cause (εἰκῇ) is condemned; but if anyone should get angry for good reason, either by way of chastisement or out of spiritual zeal he is not condemned. For even Paul spoke words of anger to Elymas the Magician and to the high priest, not without good cause, but out of zeal (Acts 13:6-12; 23:2-3). But when we get angry over money or opinions, then it is without good cause.*
Regarding which is the "correct" text, one must first decide what makes one version more "correct" than another. Antiquity is a poor measure of correctness: the oldest manuscript of the Book of Revelation, for example, lists the number of the beast as 616 and not 666. The presence or absence of some phrase in the oldest manuscript we have on hand does not preclude the phrase from having been absent or present in some still older manuscript that has been lost.
The Chalcedonian Eastern Orthodox Church, which retains the use of the Greek Scriptures, has chosen to retain without good cause in most translations of Matthew as being reflective of what was handed down by the Apostles. It can be found, for example, in the Greek Patriarchal Text of 1904. This can also be seen in the sampling of Eastern Orthodox texts that were used to compile the "Textus Receptus" in the 16th century. The pre-schism western Church, however, followed a tradition of omitting the adverb - as can be seen in what has come down in the Vulgate:
Ego autem dico vobis: quia omnis qui irascitur fratri suo, reus erit judicio.
But I say to you, that whosoever is angry with his brother, shall be in danger of the judgment (Douay-Rheims)
* Explanation of the Gospel According to St. Matthew (tr. from Greek; Chrysostom Press, 1992), p.50