Romans 10:6 is not an easy text to understand. Here it is in context:

Rom. 10:1 Brothers and sisters, my heart?s desire and prayer to God for them is that they may be saved. 2 I can testify that they have a zeal for God, but it is not enlightened. 3 For, being ignorant of the righteousness that comes from God, and seeking to establish their own, they have not submitted to God?s righteousness. 4 For Christ is the end of the law so that there may be righteousness for everyone who believes.
Rom. 10:5 Moses writes concerning the righteousness that comes from the law, that ?the person who does these things will live by them.? 6 But the righteousness that comes from faith says, ?Do not say in your heart, ?Who will ascend into heaven??? (that is, to bring Christ down) 7 ?or ?Who will descend into the abyss??? (that is, to bring Christ up from the dead). 8 But what does it say?
?The word is near you,
on your lips and in your heart?
(that is, the word of faith that we proclaim); 9 because if you confess with your lips that Jesus is Lord and believe in your heart that God raised him from the dead, you will be saved. 10 For one believes with the heart and so is justified, and one confesses with the mouth and so is saved. 11 The scripture says, ?No one who believes in him will be put to shame.? 12 For there is no distinction between Jew and Greek; the same Lord is Lord of all and is generous to all who call on him. 13 For, ?Everyone who calls on the name of the Lord shall be saved.?

Paul here is quoting Deuteronomy 9:4 and that text clarifies our “heaven” text. So, here it is:

When the LORD your God thrusts them out before you, do not say to yourself, ?It is because of my righteousness that the LORD has brought me in to occupy this land?; it is rather because of the wickedness of these nations that the LORD is dispossessing them before you. 5 It is not because of your righteousness or the uprightness of your heart that you are going in to occupy their land; but because of the wickedness of these nations the LORD your God is dispossessing them before you, in order to fulfill the promise that the LORD made on oath to your ancestors, to Abraham, to Isaac, and to Jacob.6 Know, then, that the LORD your God is not giving you this good land to occupy because of your righteousness; for you are a stubborn people.

First, the resonance of Deut 9:4 in Rom 10:6 means that Paul wants his readers to understand that the righteousness of God comes from faith in what God can do and has done and not in what humans have done and can do.
And then Paul quotes from Deut 30:12: “It is not in heaven, that you should say, ?Who will go up to heaven for us, and get it for us so that we may hear it and observe it?? And that context teaches that God’s word is near to people, not far off in heaven or beyond the sea.
Second, then, heaven in Romans 10:6 refers to the far off unreachable place.
Third, this means then that what Paul is talking about is this: humans are not to climb to the heavens with their doing nor are they to descend into the depths of powerful efforts, for both actions dethrone the very thing Christ has done for us. Instead, humans are to confess Christ and to trust in what God does in Christ. It is about a heart relationship with God. This heart relationship is made possible because Christ is now made known to both Jews and Gentiles in Christ.
Heaven there then basically refers, negatively, to something out of reach.