First, a distinction should be made about “heaven”,
There is the Intermediate heaven which is a spiritual place-not part of this creation (Heb 9:11, 24); it is the abode of God (Psalms 53:2); where Jesus currently sits at the right hand of the Father (Rom 8:34); where the consciousness of deceased Christians go to be with the Lord until He returns (2Cor 6:9, 1Thes 5:10). This however is only an intermediate state.
The final promise of eternal life as given in the Bible and from the Apostles is for bodily resurrection. Jesus is coming back to earth. When he does the dead Believers will literally and bodily rise from the dust of the grave to live forever upon a restored/renewed earth. Thus ultimately for eternity “heaven” is actually upon earth. This is the ETERNAL state.
So the confusion is that when we speak about “heaven” we really need to qualify which state of heaven is being referenced. All Christian denominations believe in bodily resurrection. It is a mandatory doctrine. According to Paul, without Resurrection, Christianity is a lie and a sham (1Cor 15). Unfortunately, however many Christians have been influenced by Platonism and ignore the resurrection (although they probably pay lip service to it).
Now to answer your question for both states, eternal and intermediate
ETERNAL heaven- of course we will know, or at least have access to the knowledge, about what is going on in earth, because we will ourselves be on earth. I don’t think we will ever become omniscient like God, we will always be learning. So we won’t know everything about things on earth. But since eternal heaven is upon the restored earth, such knowledge will certainly be within our purview.
INTERMEDIATE heaven- This is probably what you meant with the original question. It is trickier, since the Bible doesn’t directly address the topic. However we can glean some tidbits. In Rev 6:9-11 the souls of saints under the heavenly altar ask God when He will judge the earth and avenge their slain blood. This seems to indicate that the dead in Intermediate heaven have some memory of earth.
Phil 1:23-24 speaks of departing to be with the Lord is nicer, but to remain in the body is needed to help people. This strongly implies that dead saints in the Intermediate heaven have ineffectual knowledge or power about earth.
2Cor 5:6-9 Paul speaks of being “away from the body and being with the Lord.” Again, implying that he not of much use to anybody when dead.
Heb 12:1 speaks of a “cloud of witnesses that surround us.” Some people have interpreted this to mean that dead saints are watching us. However that interpretation is out of context. Verse 12:1 is referencing back to the list of faithful men given in chapter 11, that chapter is sometimes called the Heroes of Faith. These witnesses (martyr in Greek) are not witnesses of us, as if they are observing our behavior; they are witnesses of Christ, examples for us.
I believe that dead saints in a spiritual heaven have only a foggy, vague understanding of events on earth. With our physically bound minds, we envision dead saints in heaven leaning over the edge of a cloud and viewing our actions and behavior. Catholics pray to Mary and other saints, thinking that they may be able to intercede for us. Currently our minds and bodies are very much stuck in this physical realm. Paul wrote in 1 Cor 13:12 (KJV), “For now we see through a glass, darkly.” The word “glass” in the KJV actually means “mirror.” The mirrors of the ancient world were not made of glass, as today, but of metal. Bronze mirrors have been discovered in Egyptian tombs dating back to 2000 B.C. The Roman period mirror was commonly about six inches in diameter, shiny bronze with a long handle, very similar to ladies’ vanity mirrors still available today. Mirrors of antiquity did not give a clear reflection. The image was similar to what you would get from looking into the chrome bumper of your car.
What Paul means is that we perceive spiritual things as through the distorted view of a bad mirror. Humans have an imperfect perception of reality. Our understanding of our own dimension is fairly good. Our dimension is the real, tangible universe. We live in a three-dimensional space of the world. This is what we can hope to know. Spiritual things are, to some degree, beyond our comprehension.
This is speculative, but I think that the situation is somewhat inversely similar for the spirits of righteous people who abide currently in the intermediate heaven. I think that the spirits/souls of dead saints are incomplete. The Bible says that they yearn to be clothed with a resurrected body, not to remain unclothed as disembodied spirit (2Cor 5:1-4).
Can people in Heaven really know what is happening on earth? A lot of people wonder about that. The answer is: absolutely, people in Heaven do know what is happening on earth and under the earth.