Do I have to ask Jesus into my heart to be born again?

"From the first day that you set your heart to understand I have heard you.", Dan. 10:12.  John 3:5.

Not in so many words, but you do have to allow Him into your life, and live according to His word.

No, you don't.  You need to exercise faith in Jesus Christ (sufficient to motivate you to find things out for yourself); repent of whatever sins you might have (regret them, give them up, make what amends you can, etc.); be baptized by immersion by someone with authority from Christ (not some book or seminary) to do so; and be confirmed and receive the gift of the Holy Ghost.  Followed by a lifetime of Christian discipleship.  Being saved is not a single act, but a lifelong (sometimes longer) process.

Nowhere does the Bible say to "ask Jesus into your heart" in order to be saved. The truth is that death leads to heaven or hell, and the only way to heaven is through faith in Jesus Christ.

The Bible is very clear that no one can earn their way to heaven since good works do not pay for sins. See Gal. 2:16, Rom. 3:23-28, 4:4-5, Eph. 2:8-9, Tit. 3:5. The only way to heaven is by believing in Jesus for His free gift of eternal life. Jesus shed His blood on the cross as the full payment for our sins. He did all the work; all He requires of us is that we believe in Him. Once you believe in Jesus Christ to save you, you are secure forever!

Because we have all sinned against a perfectly holy and righteous God, we all deserve eternal separation from God in hell (Rom. 6:23). However, God loves us so much that He sent His only Son Jesus to die on the cross as our Substitute so we don't have to go to hell. Christ died for our sins, was buried, and rose from the grave three days later (1 Cor. 15:3-4). He gives eternal life freely to anyone who simply believes in Him for it. No one can be saved by doing good works. Salvation is by grace alone through faith alone in Christ alone (Eph. 2:8-9).

"Verily, verily, I say unto you, He that believeth on me hath everlasting life" (John 6:47).

Baptism with water, according to the accepted, trinitarian formula is best.

All Christians are born again in baptism.

Putting Jesus' teachings in your heart is a real benefit for survival but to be born again it's up to God's choosing. These are bought or chosen from mankind by God. 

Chosen to be born again to rule over the earth as kings and priests.

(1 Peter 2:9) But you are “a chosen race, a royal priesthood, a holy nation, a people for special possession,

(Revelation 5:10) and you made them to be a kingdom and priests to our God, and they are to rule as kings over the earth.”

(Revelation 20:6) Happy and holy is anyone having part in the first resurrection; over these the second death has no authority, but they will be priests of God and of the Christ, and they will rule as kings with him for the 1,000 years.

The other sheep with an earthly hope and subjects of God's kingdom.

(Psalm 37:10, 11) Just a little while longer, and the wicked will be no more; You will look at where they were, And they will not be there. But the meek will possess the earth, And they will find exquisite delight in the abundance of peace.

(Psalm 37:29) The righteous will possess the earth, And they will live forever on it.(Matthew 5:5) “Happy are the mild-tempered, since they will inherit the earth.

The answer you get will depend on whether someone subscribes to orthodox Christianity, or Arminianism/Pelagianism.

Orthodox Christianity says no.

Arminianism/Pelagianism says yes.

The belief turns on who moves first.

Christianity teaches that God moves and man responds.

Arminianism teaches man moves first, and God responds.

This information is available to anyone willing to explore a little historical theology 

No.

Joh 3:3  Jesus answered him, “Truly, truly, I say to you, unless one is born again he cannot see the kingdom of God.” 

Joh 3:4  Nicodemus said to him, “How can a man be born when he is old? Can he enter a second time into his mother's womb and be born?” 

Joh 3:5  Jesus answered, “Truly, truly, I say to you, unless one is born of water and the Spirit, he cannot enter the kingdom of God.

Note there is nothing here about asking Jesus into your heart.  In fact, you'll find nothing in the Bible about asking Jesus into anyone's heart.  When it comes to the heart, here's what IS written.

Act 2:37  Now when they heard this they were cut to the heart, and said to Peter and the rest of the apostles, “Brothers, what shall we do?” 

Act 2:38  And Peter said to them, “Repent and be baptized every one of you in the name of Jesus Christ for the forgiveness of your sins, and you will receive the gift of the Holy Spirit. 

Yes, but there is more to it than just that. You should read one of the Gospels (or all) in order to grasp the real meaning of being 'born again'.