At the rapture, when all believers are taken out, does that mean the Holy Spirit is gone too?
No. When believers are taken out, the Holy Spirit finishes His role as indwelling individual believers, but that does not remove Him from the earth. He remains the omnipresent Spirit of God, so people can still be saved during the tribulation.
Les explains that the question is whether the rapture removes the Holy Spirit because He indwells believers. His answer is that just because the Holy Spirit goes out and finishes His role as indwelling the individual believer, that does not remove Him from the earth. The Holy Spirit has always been the omnipresent of God on the planet. He moved upon the face of the deep in Genesis 1 and was evident through the Old Testament, though He did not indwell Old Testament believers like He does today. Likewise in the tribulation, the indwelling person of the Godhead will end, but He will remain on the earth as the omnipresent Spirit of God; otherwise no one could be saved during the tribulation, and we know they will be. Every believer now in this age of grace of which Paul is dealing is the temple of the Holy Spirit.