In the Old Testament days, the Holy Spirit did not indwell the faithful, it came upon them. Sometimes the Spirit left them. John explains that until Pentecost, the Spirit had not been given, since Jesus had not yet been glorified. (John 7:39) It wasn't until Acts that the Spirit began to indwell believers. (Acts 1:4,8; Acts 2:1-13)
In the Old Testament, the full mystery of the Trinity was not revealed. Some clues show us that occasionally the OT Saints knew of a coming Redeemer, and that they knew of the Spirit, but not that there was Father-Son-Holy Spirit working together and as One God.
That was the past. In the future, the Tribulation Saints will not receive the Spirit as an indwelling deposit of guaranteed eternal security, either. The Spirit will return to the OT ministry of drawing the people to God and encouraging them from the outside of the body. 2 Thessalonians 2 overall has a good NT interpretation of the Spirit's work during the Tribulation. In sum, the ministry of the indwelling Holy Spirit is a gift to the Church Age saints and to us alone.
Why do the Church Age Saints receive this gift, but not Old Testament Saints nor Tribulation Saints? Jack Kelley thinks it is because there was plenty of evidence of God's work in the world during Old Testament times and there will be again in Tribulation times, but the Church Saints are told to believe by faith. (John 20:29). It is a hard go to believe during the church age, because although God has made Himself evident in the world so that none would be without excuse, (Romans 1:19-20) satan is god of this world (2 Cor 4:4) and does his deceiving work constantly, blinding the minds of all those who do not have the spirit of the Lord within them.
In the Tribulation, just as it was in the OT, the saints will be responsible keeping their faith alive without indwelling support of the Holy Spirit. There are many verses which urge the Tribulation saints to overcome, to persevere. Revelation 12:14 is one, "Here is the perseverance of the saints who keep the commandments of God and their faith in Jesus." Jesus said that of the last days, "And ye shall be hated of all men for my name's sake: but he that endureth to the end shall be saved." Matthew 10:22 and also Matthew 24:13 and Revelation 2:26-28 urge perseverance. A vivid illustration of lack of perseverance is from Mark 4:17: "And have no root in themselves, and so endure but for a time: afterward, when affliction or persecution ariseth for the word's sake, immediately they are offended."
There is precedent for God's removal of the Spirit prior to a worldwide judgment. In Genesis 6:3 He says, "My Spirit will not strive with man forever, because he also is flesh; nevertheless his days shall be one hundred and twenty years." The Great Tribulation will not start until He is taken out of the way (but not entirely removed), as 2 Thessalonians 2:7 states. Genesis 19:22 also shows us a picture of the Tribulation as Lot is standing at the edge of the city and dallying and the angel tells him he cannot start the judgment "until you are safely out of the way." The Spirit will not be indwelling believers during the Tribulation! Imagine your own faith during this time of the Church Age and how hard it is already to resist sin, to grow, and to maintain devotion to the Lord through trials. Now imagine how hard it will be for Tribulation Saints, and then worse, their eternal security is NOT assured! Knowing this, every Christian should be on their knees, every night, praying for the lost, and every day, rustling the hedges for unbelievers to share the Gospel with!
During the Tribulation it will be extraordinarily hard for people to come to faith. Even though they know Who is bringing the plagues and judgments on them, as Revelation 6:21 shows us: "And huge hailstones, about one hundred pounds each, came down from heaven upon men; and men blasphemed God because of the plague of the hail, because its plague was extremely severe," they still refuse to repent. They simply refuse:
"And the rest of mankind, who were not killed by these plagues, did not repent of the works of their hands, so as not to worship demons, and the idols of gold and of silver and of brass and of stone and of wood, which can neither see nor hear nor walk; and they did not repent of their murders nor of their sorceries nor of their immorality nor of their thefts." (Revelation 9:20,21)
The Church Age is full of grace and mercy, and that is thanks to the gift of the indwelling Holy Spirit. If you are praying for someone to come to repentance of their sins and to faith in the Resurrected Jesus, pray that they do so now. Don't say "well, at least during the Tribulation they will SEE God's hand at work and come to believe, even though it will be terrible." The Spirit will not be there to help them as an indwelling ministry, hearts will be hardening daily, and even if they do come to faith, they are at risk of losing it, like Saul did. If you are one who is on the fence and figuring that the rapture will be the sign and you'll repent then and afterward, the same is true for you. "Behold, now is the accepted time; behold, now is the day of salvation" (2 Corinthians 6:2).