In reading Albert Barnes' Notes on I Timothy, he talks about genealogies which are referring to the verse I Timothy 1:4 "nor to devote themselves to myths and endless genealogies, which promote speculations rather than the stewardship from God that is by faith."
Albert Barnes states that the Hebrews kept careful genealogical records because it was necessary in order that the distinction of their tribes might be kept up. What was the importance of keeping up the distinction of their tribes, and their blood and of birth? Where does Albert Barnes come to the conclusion that the Hebrews kept such records? In other words, what other verses or other information would be beneficial in making his point? And what does Paul mean in I Timothy 1:4 when he says "...endless genealogies, which promote speculations rather than the stewardship from God that is by faith".
"As I urged you when I went into Macedonia -- remain in Ephesus that you may charge some that they teach no other doctrine, nor give heed to fables and endless genealogies, which cause disputes rather than godly edification which is in faith" (I Timothy 1:3-4).
"But avoid foolish disputes, genealogies, contentions, and strivings about the law; for they are unprofitable and useless" (Titus 3:9).
One common trait of false doctrines is that the proponents often major in minor matters. The warning, both to Timothy and Titus, is not to get caught up in arguments over silly things. The Jews were famous for this, spending countless hours arguing over minor details. Jesus scolds them thus, "Woe to you, scribes and Pharisees, hypocrites! For you pay tithe of mint and anise and cummin, and have neglected the weightier matters of the law: justice and mercy and faith. These you ought to have done, without leaving the others undone" (Matthew 23:23). It isn't that small details are unimportant, but that they should not dominate our thoughts.
There are two main thoughts concerning genealogies. Some believe that Paul is referring to arguments about personal genealogies. Adam Clarke makes mention of this in his commentary, too. "We are told that Herod destroyed the public registers; he, being an Idumean, was jealous of the noble origin of the Jews; and, that none might be able to reproach him with his descent, be ordered the genealogical tables, which were kept among the archives in the temple, to be burnt. See Euseb. H. E., lib. i. cap. 8. From this time the Jews could refer to their genealogies only from memory, or from those imperfect tables which had been preserved in private hands; and to make out any regular line from these must have been endless and uncertain"
One of the false teachings that sprang up during this era was Gnosticism. It was based on a belief that there was secret knowledge not available to most people (Gnostic means "knowledge"). One aspect of this was the use of genealogies to prove various ideas. Some commentators believe that Paul is alluding to the early disputes that gave rise to Gnosticism. Jamieson, Faucett, and Brown state concerning these genealogies, "not merely such civil genealogies as were common among the Jews, whereby they traced their descent from the patriarchs, to which Paul would not object, and which he would not as here class with "fables," but Gnostic genealogies of spirits and aeons, as they called them, "Lists of Gnostic emanations" [ALFORD]. So TERTULLIAN [Against Valentinian, c. 3], and IRENAEUS [Preface]. The Judaizers here alluded to, while maintaining the perpetual obligation of the Mosaic law, joined with it a theosophic ascetic tendency, pretending to see in it mysteries deeper than others could see. The seeds, not the full-grown Gnosticism of the post-apostolic age, then existed. This formed the transition stage between Judaism and Gnosticism. "Endless" refers to the tedious unprofitableness of their lengthy genealogies (compare Tit 3:9)."
Of the latter, we see this continuing today. The DaVinci Code was the latest of a series of Gnostic influenced books that tries to establish goddess worship founded in part on mythological genealogies.
Another takes place in Messianic Judaism. I've heard of men telling people that they are really Jews because of their names and then telling them that there two different covenants in effect. Those who are Jewish are supposed to keep the Mosaical Law and the rest are suppose to keep Christ's Law.
Whether we are talking about Jewish genealogical tables or imaginary lineages, the point remains the same -- some topics are not worth the breath to discuss.
"But reject profane and old wives' fables, and exercise yourself toward godliness" (I Timothy 4:7).
"If anyone teaches otherwise and does not consent to wholesome words, even the words of our Lord Jesus Christ, and to the doctrine which accords with godliness, he is proud, knowing nothing, but is obsessed with disputes and arguments over words, from which come envy, strife, reviling, evil suspicions, useless wranglings of men of corrupt minds and destitute of the truth, who suppose that godliness is a means of gain. From such withdraw yourself" (I Timothy 6:3-5).
"O Timothy! Guard what was committed to your trust, avoiding the profane and idle babblings and contradictions of what is falsely called knowledge -- by professing it some have strayed concerning the faith. Grace be with you. Amen" (I Timothy 6:20-21).
"Remind them of these things, charging them before the Lord not to strive about words to no profit, to the ruin of the hearers. Be diligent to present yourself approved to God, a worker who does not need to be ashamed, rightly dividing the word of truth. But shun profane and idle babblings, for they will increase to more ungodliness. And their message will spread like cancer. Hymenaeus and Philetus are of this sort, who have strayed concerning the truth, saying that the resurrection is already past; and they overthrow the faith of some" (II Timothy 2:14-18).
"But avoid foolish and ignorant disputes, knowing that they generate strife" (II Timothy 2:23).
"For the time will come when they will not endure sound doctrine, but according to their own desires, because they have itching ears, they will heap up for themselves teachers; and they will turn their ears away from the truth, and be turned aside to fables" (II Timothy 4:3-4).
"This testimony is true. Therefore rebuke them sharply, that they may be sound in the faith, not giving heed to Jewish fables and commandments of men who turn from the truth" (Titus 1:13-14).