Christians often say; after the sin fall God can’t have a direct relation with human. What are the teachings that this is not the case?
Firstly, God is perfect, and there is nothing that He "can"t" do. Secondly, true, most of us haven't received direct messages from God, but throughout the Tanach there are hundreds of prophets and prophecies where God spoke to man, that contradict that Christian theory (e.g. Sam. I, 10, 5), not to mention our cleaving to God and having Him address the entire Jewish nation at Sinai (Dvarim 4). In addition, we speak to our Father in heaven all the time in prayer, whether formal or informal, and so did our fore-fathers throughout the Tanach, especially seen and felt throughout Tehillim (Psalms). True it seems difficult, because our human relationships are with people whom we can see, and are based upon, and grow through, mutual giving, and God does not lack anything for us to give to Him, but this was the case also before man sinned in the Garden of Eden, and nothing has changed. Nevertheless, He tells us that we can "give" Him sacrifices and good deeds, which helps us feel we are giving (to strengthen our relationship with Him from our point of view), but it's really for our sake. Similar to a parent who "excitedly" accepts a primitive drawing from his 4 year-old child, and even hangs it on the refrigerator, even though it's not really a "masterpiece", but it helps the child feel good and strengthens the relationship. So too God gives us commandments which are really for our benefit to improve ourselves and to make us more Godly, but they also aid us in developing our relationship with Him. Christians have a problem, because they did away with almost all of those 613 practical mitzvot, but religious Jews do feel a direct relationship with God, and speak with Him 3 times a day on a regular basis.