What does it mean to be a living sacrifice? Sacrifices made to God were for the most part sacrifices of live animals, offered up to be killed and their bodies consumed by fire on the altar of sacrifice,
"And you shall burn the whole ram on the altar. It is a burnt offering to the LORD; it is a sweet aroma, an offering made by fire to the LORD". Exodus 29:18
There were precise and exacting instructions as to how the offering was to be offered, then prepared to be sacrificed, and then to be burned on the altar. The act was an act of worship that came up to God as a sweet aroma made by fire.
Fire is almost always a metaphor for the work and presence of the Holy Spirit. Fire is a force of purification, and endowment of power to speak and proclaim the Word of God. Isaiah experienced this way in Isaiah 6:1-8
"In the year that king Uzziah died I saw also the Lord sitting upon a throne, high and lifted up, and his train filled the temple. 2 Above it stood the seraphims: each one had six wings; with twain he covered his face, and with twain he covered his feet, and with twain he did fly. 3 And one cried unto another, and said, Holy, holy, holy, is the LORD of hosts: the whole earth is full of his glory. 4 And the posts of the door moved at the voice of him that cried, and the house was filled with smoke.
Then said I, Woe is me! for I am undone; because I am a man of unclean lips, and I dwell in the midst of a people of unclean lips: for mine eyes have seen the King, the LORD of hosts. 6 Then flew one of the seraphims unto me, having a live coal in his hand, which he had taken with the tongs from off the altar: 7 And he laid it upon my mouth, and said, Lo, this hath touched thy lips; and thine iniquity is taken away, and thy sin purged. 8 Also I heard the voice of the Lord, saying, Whom shall I send, and who will go for us? Then said I, Here am I; send me." Isaiah 6:1-8
Now we are called to present our bodies as living sacrifices. But how can you be both a sacrifice which forfeits its life, and living at the same time?
By becoming crucified with Christ, then by being raised up with Him to sit in Heavenly places.
"I have been crucified with Christ; it is no longer I who live, but Christ lives in me; and the life which I now live in the flesh I live by faith in the Son of God, who loved me and gave Himself for me." Galatians 2:20
"And those who are Christ’s have crucified the flesh with its passions and desires". Galatians 5:24
Being crucified with Christ means my flesh and all its human desires are dead as far as who I am now as one alive in Christ. And as God has raised up Jesus from the dead, He has also raised us up with Him to live in holiness and have eternal life with Him in Heaven.
"But God, who is rich in mercy, for his great love wherewith he loved us, Even when we were dead in sins, hath quickened us together with Christ, (by grace ye are saved;) And hath raised us up together, and made us sit together in heavenly places in Christ Jesus: That in the ages to come he might shew the exceeding riches of his grace in his kindness toward us through Christ Jesus. For by grace are ye saved through faith; and that not of yourselves: it is the gift of God: Not of works, lest any man should boast." Ephesians 2:4-9
As living sacrifices, we make our whole purpose and reason for being alive to serve the cause and purposes of God, to glorify Him in all things and take no thought for ourselves, especially, to satisfy the dictates of the flesh, but rather to resist the devil in all his temptations and he will flee from us, and give ourselves whole heartedly to the ministry of God's higher calling in Christ Jesus.
This does not mean that we are to be myopically concentrating on mustering up some super spiritual pseudo-holy posture 24/7, but rather that we make ourselves available to the leading of the Holy Spirit to act or speak whatever He prompts in our hearts to do or say at any time.