The answer is no. Nonbelievers do not welcome or understand the Holy Spirit’s ways and teachings (1 Cor. 2:14). The Bible tells us that nonbelievers are spiritually dead (Eph. 2:1–3), spiritually blind (2 Cor. 4:3–4), and are separated from God (Eph. 4:17–19). They are under sin’s ruling power (Rom. 3:9). The Bible describes nonbelievers as God’s enemies (Rom. 5:10), hostile to God (Col. 1:21), and incapable of pleasing Him (Rom. 8:5–8). They are unable to rescue themselves from their lost condition (Rom. 5:8). The Spirit of God is not resident in lost people; indeed, His absence is evidence that they need the Savior (John 14:16–17; Rom. 8:9).
This is not to say that the Spirit is not concerned about lost people. He is. Through the proclamation of the good news, the Spirit speaks to the lost (1 Peter 1:12), convicts them of their sin, and opens their minds to understand and see their lost condition (Acts 2:37). Thus convicted, some unsaved people come to faith in Christ (Acts 16:14).
The good news is that when we receive Christ as our personal Lord and Savior under the convicting work of the Spirit (John 16:7–9; 2 Thess. 2:13; 1 Peter 1:2), we are sealed (Eph. 1:13–14), anointed (2 Cor. 1:21; 1 John 2:20, 27), and baptized by the Holy Spirit into Christ’s body (1 Cor. 12:12–13). A new believer is regenerated (John 3:1–8; Titus 3:4–6) and indwelt by the Spirit (1 Cor. 6:19).
The indwelling presence of God the Holy Spirit in a person’s heart and spirit is evidence of a person’s eternal relationship with God through faith in Christ (Eph. 1:14). The Spirit of God will never leave a believer but will reside in the believer forever. His indwelling presence is an incalculable blessing for us, both in this life and the life to come (John 14:16).
More: Can a believer lose the Spirit of God?
This is a complex question. That being said, my assumption would be that the person was never actually saved. The person who renounces Jesus, according to Hebrews 6:1–8, was never in relationship with God in the first place. They are called “apostates” or “antichrists” (see 1 John 2:18–20). In such a circumstance, the Spirit did not leave because the Spirit never was in the life of this person. If this person was a believer, the Holy Spirit would not leave, but the Bible tells us we can grieve (Eph. 4:30) or quench the Spirit’s power (1 Thess. 5:19) even as believers.