Jesus has many titles in scripture including The Word of God.
John 1:1 In the beginning was the Word, and the Word was with God, and the Word was God.
John 1:14 And the Word was made flesh, and dwelt among us, (and we beheld his glory, the glory as of the only begotten of the Father,) full of grace and truth.
1 John 1:1 That which was from the beginning, which we have heard, which we have seen with our eyes, which we have looked upon, and our hands have handled, of the Word of life;
1 John 5:7 For there are three that bear record in heaven, the Father, the Word, and the Holy Ghost: and these three are one.
Revelation 19:13 And he was clothed with a vesture dipped in blood: and his name is called The Word of God.
One title people use refer to Jesus is “the Living Word”.
There is nothing necessarily wrong with referring to Him using this title, Jesus is the Word made flesh, He is the “Word of Life”, and Jesus was, and is, and ever will be indeed alive and living.
However, this title is not one He has used in scripture to identify Himself. So, while referring to Jesus as the “Living Word” is not necessarily wrong, it is unwise to build a doctrine off of that title being that he has not used that as a title for Himself in scripture.
Some folks will say things like “we don’t need a perfect written word because we have Jesus, The Living Word”. This is a poor assumption based on something that is not in scripture.
On the other hand, we do see verses referring to the written word of God as living:
Hebrews 4:12 For the word of God is quick, and powerful, and sharper than any twoedged sword, piercing even to the dividing asunder of soul and spirit, and of the joints and marrow, and is a discerner of the thoughts and intents of the heart.
The context here is scripture, the written word of God. Here we see that the word of God is “quick”, meaning alive, life-giving, living.
Also in 1 Peter
1 Peter 1:23 being born again, not of corruptible seed, but of incorruptible, by the word of God, which liveth and abideth for ever.
Here again we see a reference to the written word, saying the word of God liveth (living).
So while the bible does not have a verse calling Jesus the “Living Word”, we do see verses saying God’s words are living.