In John 3:16, the Greek word translated "believes" is πιστευων, pisteuon in our alphabet. This a present tense form of the Greek verb πιστευω, pisteuo in our alphabet, word number 4100 in Strong's Greek Dictionary. The present tense does not even imply continuous belief. Such an implication would require the aorist tense, which was NOY used here.
Again, the same form of the same Greek word is translated "believes" in John 5:24. And in that verse, we are explicitly told that such a person "has passed from death into life." The Greek word translated "has passed" is μεταβεβηκεν, metabebeken in our alphabet, a past perfect form of the Greek verb μεταβαίνω, metabaino in our alphabet, word number 3327 in Strong's Greek Dictionary. The past perfect tense not only states that the action took place in the past, but that it was completed in the past. So this word translates literally as "has already passed," as some modern ranslations render it.
So, not only is no action on our part required for salvation, the scriptures do not even imply that a continuous belief is required. Once the belief is real, an absolute and irrevocable (see Romans 11:29) change has already taken place.
As for John 5:24:
In John 5:24 it says that we have to not only believe in Jesus, but we have to hear His word (obey) Jesus so as to pass from life unto death. For John 12:48 says that he (the person) who does not receive His words, those words will judge them on the last day. What words of Jesus can we not receive that can judge us? Oh, things like Matthew 5:28-30, Matthew 6:15, Matthew 12:37, Matthew 25:31-46, Luke 9:62 just to name a few. So we have to be in constant state of belief and obedience to Jesus in order for the promise of John 5:24 to be applied. Also, Jesus alone has immortality (1 Timothy 6:16). 1 John 5:12 says he that has the Son has life and he that does not have the Son does not have life. So we need to abide in Christ who is the source of eternal life. One of the ways we can have an assurance that we know Christ is if we find that we are keeping His commandments (1 John 2:3). But the person who says they know the Lord and they do not keep His commandments, they are a liar and the truth is not in them (1 John 2:4). What truth is not in them? Jesus. For Jesus is the way, the truth, and the life (John 14:6).
As for Romans 11:29:
Well, when you read Romans 11:29, you have to read it in context. Romans 9-11 deals with Israel and their lack of not accepting the Messiah and His grace. The one time free will offer of accepting Jesus and His grace and His call will not be taken away for Israel. God will not take this new way of salvation away for them if they change and repent and accept Jesus. That free will offer is still on the table for Israel. For we know Romans 11:29 is talking to the unsaved Jew because Romans 11:28 in context says this: “As concerning the gospel,
they are enemies for your sakes: but as touching the election,
they are beloved for the fathers' sakes.”
Also, Romans 11 is proof against Eternal Security because it says this:
20 “Well; because of unbelief they were broken off, and thou standest by faith. Be not highminded, but fear:
21 For if God spared not the natural branches, take heed lest he also spare not thee.
22 Behold therefore the goodness and severity of God: on them which fell, severity; but toward thee, goodness, if thou continue in his goodness: otherwise thou also shalt be cut off.” (Romans 11:20-22).
It says that the Jews were broken off because of unbelief. No eternal security saved them. It says that we stand by faith and we are told to be not high minded but to fear. Why? It says if God spared not the natural branches, take heed that He also spare not you. Thus we have to continue in God’s goodness otherwise we can be cut off, too. What goodness? A sin and still be saved type belief? No. God’s goodness can be His grace if we do honestly stumble on rare occasion unintentionally, but His goodness is actually living holy after we are saved by His grace. For Titus 2:11-12 says that God’s grace teaches us to deny ungodliness and that we should live righteously and godly in this present world.