Does the Holy Spirit leave the earth when believers are taken out at the rapture?
No. At the rapture, the Holy Spirit ceases His role of indwelling individual believers, but He does not leave the earth. He remains the omnipresent Spirit of God on the planet, just as He was in the Old Testament and will be during the tribulation so people can be saved.
Les explains that the question arises because when all believers are taken out at the rapture, some wonder whether the Holy Spirit is gone too. His answer is that the Holy Spirit goes out and finishes His role as indwelling the individual believer, but that does not remove Him from the earth. Les points out that the Holy Spirit has always been the omnipresent of God on the planet, going back to Genesis chapter 1, where the Spirit moved upon the face of the deep. In the Old Testament the Holy Spirit was evident, though He did not indwell Old Testament believers as He does today. Likewise in the tribulation, His indwelling role 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.