Many passages from the Bible have been used as the basis for people seeing and hearing from heaven. I think that I have now rounded up all the major points. I could be wrong, perhaps there are even more. Below I give the 7 major biblical points that I have seen brought up as supporting this view. Some of these points are related, but they all approach the issue from a slightly different perspective.
One important thing to remember is that the dead in heaven might not "see and hear" exactly like we do. This does not mean that they can't see and hear, but only that the faithful in heaven might see and hear things differently.
1: Faith in Heaven
20 He said to them, “Because of your little faith. Amen, I say to you, if you have faith the size of a mustard seed, you will say to this mountain, ‘Move from here to there,’ and it will move. Nothing will be impossible for you.”
Many things are impossible for us of little faith, but I would guess that the dead in heaven have faith at least the size of a mustard seed. If they do, then it is possible for them to hear us. (Of course, this passage does not say they must listen, but there is nothing in the Bible that suggests that they would not listen to people who cry out to them, or that God would command them not to listen when a loved one wants them to hear.)
2: The Psalms
Psalms are prayers. In at least two Psalms, the person who is saying these prayers is asking or commanding all of creation (including those in heaven, angels, hosts, and so on) to praise God:
20 Bless the LORD, all you his angels,
mighty in strength, acting at his behest,
obedient to his command.
21 Bless the LORD, all you his hosts,
his ministers who carry out his will.
1 Hallelujah! Praise the LORD from the heavens; praise him in the heights.
2 Praise him, all you his angels; give praise, all you his hosts.
Daniel 3:86 (not available in some Bibles)
86 Spirits and souls of the just, bless the Lord;
praise and exalt him above all forever.
Some might say that these passages are just figurative expressions of the psalmist's own praise. However:
13 Then I heard every creature in heaven and on earth and under the earth and in the sea, everything in the universe, cry out:
“To the one who sits on the throne and to the Lamb
be blessing and honor, glory and might,
forever and ever.”
39 Some of the Pharisees in the crowd said to him, “Teacher, rebuke your disciples.”
40 He said in reply, “I tell you, if they keep silent, the stones will cry out!”
8 Take the staff and assemble the community, you and Aaron your brother, and in their presence command the rock to yield its waters. Thereby you will bring forth water from the rock for them, and supply the community and their livestock with water.
So according to God (literally the words of God in this case), it is even appropriate to speak directly to rocks and expect something to happen. Of course, rocks are inanimate, which means they have no soul. But people (even dead people, of course) have souls, so we should expect that they would hear us like people and not like rocks, even if they don't hear in exactly the same way that we do. (We know that beings with only a soul can hear, even though they have no body, because the Bible describes angels having conversations with people.)
3: Moses and Elijah
At the Transfiguration of Jesus, the Bible describes Moses and Elijah talking with Jesus as if they are aware of what is and will be happening in the world:
29 While he was praying his face changed in appearance and his clothing became dazzling white.
30 And behold, two men were conversing with him, Moses and Elijah,
31 who appeared in glory and spoke of his exodus that he was going to accomplish in Jerusalem.
Some people might say that Moses and Elijah could hear Jesus because they "came down to earth". But others say that Peter, James, and John saw a glimpse of people who were "in heaven" talking to Jesus (when Peter suggested setting up tents, a cloud overshadowed them, a voice said “This is my beloved Son. Listen to him.”, and then they could no longer see Moses and Elijah).
Revelation establishes that certain people in heaven "interact" with the prayers of other people:
8 When he took it, the four living creatures and the twenty-four elders fell down before the Lamb. Each of the elders held a harp and gold bowls filled with incense, which are the prayers of the holy ones.
This describes the prayers of holy people filling up golden bowls, in heaven. The bowls are held by the 24 elders (the 12 apostles and the 12 tribes). The elders offer up these prayers to God like incense ("Let my prayer be incense before you...", Psalm 141:2). The point here is that, at the very least, these people are directly interacting with prayers. How do people usually directly interact with prayers? By hearing them (or knowing them, or saying them).
5: Raphael's Record
Tobit 12 describes Archangel Raphael doing the same, but with more clarity: the prayers of Tobit and Sarah are recorded, and this record is something that God wants kept and "presented" to Him:
12 Now when you, Tobit, and Sarah prayed, it was I who presented the record of your prayer before the Glory of the Lord; and likewise whenever you used to bury the dead.
Obviously, God can hear all prayers, but He still wants a record kept. How did Raphael know what the prayers were? My guess is that Raphael could hear them. Of course, Raphael is an angel, not a heavenly human person, but this seems to show that this is an activity endorsed by God.
6: Abraham hearing the rich man
The following parable implies that someone in heaven is capable of hearing:
22 When the poor man died, he was carried away by angels to the bosom of Abraham. The rich man also died and was buried, 23 and from the netherworld, where he was in torment, he raised his eyes and saw Abraham far off and Lazarus at his side.
24 And he cried out, ‘Father Abraham, have pity on me. Send Lazarus to dip the tip of his finger in water and cool my tongue, for I am suffering torment in these flames.’
25 Abraham replied, ‘My child, remember that [...]
The rich man is in hell, suffering, and he cries out to Abraham. Abraham is dead and in a far-off and elevated place. Some people who argue against there being any biblical basis get this point entirely confused. It's not about the rich man seeing his family! It is about Abraham in that heavenly place hearing the rich man. Even this person in hell is heard by Abraham.
Some say that the only lesson of this parable is that people will not believe even if someone returns from the dead. Others say that the only lesson is that once you are in hell it is too late to pray, or that not all prayers will be answered, or that people who care more for riches than for the poor will go to hell. But others say that one important lesson is that faithful people like Abraham can hear prayers and are worth praying to.
The following is my own explanation of why this parable is important: There can't really be a sort of "isolation bubble" around a place like heaven, because then this parable would be describing something heretical. For example, no parable would ever describe someone being reincarnated as a cow while giving any kind of teaching, because reincarnation in general is a heresy and incompatible with the order of things. So we must believe that this parable describes something that is in accordance with the way in which God has ordered His creation.
7: Cloud of Witnesses
Hebrews is also often brought up as evidence that a "cloud of witnesses" can see us (this "cloud" refers to the dead that are in heaven, including: Abel, Enoch, Noah, Jacob, Moses, Rahab Gideon, Barak, Samson, Jephthah, David, Samuel, and the prophets, all mentioned in Hebrews 11):
1 Therefore, since we are surrounded by so great a cloud of witnesses, let us rid ourselves of every burden and sin that clings to us* and persevere in running the race that lies before us
2 while keeping our eyes fixed on Jesus, the leader and perfecter of faith. For the sake of the joy that lay before him he endured the cross, despising its shame, and has taken his seat at the right of the throne of God.
The interpretation depends on what the word "witnesses" means. Some have interpreted it to mean that the people are witnessing us (seeing us, like spectators in a stadium) as we "run the race". Others suggest that these people are witnesses to faith: good role models, a "cloud" of the faithful that we should hope surrounds us. I think the second view (that we should look up to saints) is more appropriate based on context, so this is the only point I don't entirely agree with. But I have included it because it is important.
So there are many passages from the Bible that serve as the basis for people seeing and hearing from heaven. Points 1, 3, 5, 7, and possibly 6 relate to the possibility of seeing (and not just hearing prayers).
The most important thing is to have the faith of Abraham, and to give prayer worthy to be offered up to God.