What does David mean in Psalm 51:11 when he petitions: "Do not cast me from your presence or take your Holy Spirit from me." Does this petition indicate that we can lose our salvation or the presence of the Holy Spirit in our lives? How does the Holy Spirit work in the Old and New testament anyways?

It is important to first understand the context in which this psalm is written. David is repenting over his sin with Bathsheba. In his repentance it is important to realize that the purpose of the writer is to be repentant and contrite before God, not to state theological truths about Him. Thus, it is important to read the intent and read this theological statement according to that intent.

The lines surrounding this plea are basically saying the same thing, to be restored to fellowship with God, from which his sin had separated him. It is important to understand that there is a different between the filling and indwelling of the Holy Spirit. The indwelling of the Holy Spirit is what occurs when someone is saved and given a new heart with new desires (Rom. 8:9-11; Titus 3:4-7; John 3:5). However, the Spirit also fills people to enable them to complete certain tasks. This is distinct from the regenerating work of the Spirit and is often seen in the Old Testament when kings were anointed, craftsman were prepared to make objects of the temple, and prophets were prepared to speak on behalf of the Lord (Ex. 28:3; Jdg. 3:10; 1 Sam. 16:13). The event in which the Spirit anointed David for his kingship in 1 Sam. 16:13 is the antithesis of what he pleas will not happen in verse 11. David might recall when the Spirit of the Lord departed from Samuel in 1 Sam. 16:14. Although David probably did not have this text to read, he would have most likely recognized this truth by experience or possibly from Samuel. Thus, David is pleading that the Lord would not break fellowship with him and leave him, like He did with Saul, for his sin with Bathsheba. He also pleads that the Lord would continue to have His Spirit on David so that he will remain enabled for his position of King, which is suggested in his desire to be able to teach sinners His ways (v. 13).