After thinking about your question and reading the text around that verse, I asked myself, what grieves the Holy Spirit? What does God want from us? Throughout Scripture God repeatedly says that He wants us to "know Him." Not just know of Him but to know Him intimately.
A few years ago, after listening to an audio version of the Bible, it caught my attention that I kept hearing a repeated phrase from God similar to "so they will know Me" or "so you will know that I am God." As a result of this I did a study using the NKJV and discovered at least 294 references of God declaring who He is and that He wants us to "know Him." I say "at least" because I'm still not sure I found all of them.
God wants us to know Him. Would it grieve the Holy Spirit if we ignored God and took Him for granted? God commanded that we love Him with all our heart, mind, soul, and strength. Would it grieve the Holy Spirit if we did not love God? God commands that we love each other as we love ourselves. Would it grieve the Holy spirit if we did not love each other?
In the surrounding text to Ephesians 4:30, Paul gives us many examples of what we are to do and what we are not to do. Things like putting off the old ways and putting on the new ways of God (vs. 17-24), speaking truth instead of lying, sinning in anger, giving place to the devil, not stealing...doing good, speaking good words to edify and build each other up rather than tearing each other down...putting away bitterness, wrath, anger, clamor, evil speech...being kind, tenderhearted, and forgiving toward one another (vs. 25-32). I believe all of these are examples of how not to grieve the Holy Spirit.
