In this pericope, the disciples are questioned about their personal religious practices. Jesus uses this occasion to teach a new way of thinking about personal service to God.
He draws the mind away from ceremony and points us instead to genuine selflessness.
In Luke's summary we fully realize precisely what Jesus is talking about. The people may be content to stay with the Law of Moses permanently instead of understanding its purpose in pointing us to Christ. Likewise, teaching the law of Christ without the foundation of the Law of Moses will destroy the whole effort.
The Old Testament needs the New Testament, and the New Testament needs the Old. Neither is complete without the other.
Romans 15:4
Whatever was written in former days was written for our instruction, that through endurance and through the encouragement of the Scriptures we might have hope.
Paul also draws a contrast between the ceremony of the Law and the real consequences of our choices in life.
Colossians 2:20-23
If with Christ you died to the elemental spirits of the world, why, as if you were still alive in the world, do you submit to regulations-- "Do not handle, Do not taste, Do not touch" (referring to things that all perish as they are used)--according to human precepts and teachings? These have indeed an appearance of wisdom in promoting self-made religion and asceticism and severity to the body, but they are of no value in stopping the indulgence of the flesh.
Nevertheless, Christians remain priests to our God, still drawing distinctions between the clean and the unclean. We simply do so not in terms of food or days but in terms of life choices.
1 Peter 2:4-5
As you come to him, a living stone rejected by men but in the sight of God chosen and precious, you yourselves like living stones are being built up as a spiritual house, to be a holy priesthood, to offer spiritual sacrifices acceptable to God through Jesus Christ.
Romans 12:1-2
I appeal to you therefore, brothers, by the mercies of God, to present your bodies as a living sacrifice, holy and acceptable to God, which is your spiritual worship. Do not be conformed to this world, but be transformed by the renewal of your mind, that by testing you may discern what is the will of God, what is good and acceptable and perfect.
Hebrews 5:14
Solid food is for the mature, for those who have their powers of discernment trained by constant practice to distinguish good from evil.
We also note Isaiah's teaching about fasting looms large in the background of this pericope.
Isaiah 58:1-10
Cry aloud; do not hold back;
lift up your voice like a trumpet;
declare to my people their transgression,
to the house of Jacob their sins.
Yet they seek me daily
and delight to know my ways,
as if they were a nation that did righteousness
and did not forsake the judgment of their God;
they ask of me righteous judgments;
they delight to draw near to God.
"Why have we fasted, and you see it not?
Why have we humbled ourselves, and you take no knowledge of it?"
Behold, in the day of your fast you seek your own pleasure,
and oppress all your workers.
Behold, you fast only to quarrel and to fight
and to hit with a wicked fist.
Fasting like yours this day
will not make your voice to be heard on high.
Is such the fast that I choose,
a day for a person to humble himself?
Is it to bow down his head like a reed,
and to spread sackcloth and ashes under him?
Will you call this a fast,
and a day acceptable to the LORD?
Is not this the fast that I choose:
to loose the bonds of wickedness,
to undo the straps of the yoke,
to let the oppressed go free,
and to break every yoke?
Is it not to share your bread with the hungry
and bring the homeless poor into your house;
when you see the naked, to cover him,
and not to hide yourself from your own flesh?
Then shall your light break forth like the dawn,
and your healing shall spring up speedily;
your righteousness shall go before you;
the glory of the LORD shall be your rear guard.
Then you shall call, and the LORD will answer;
you shall cry, and he will say, "Here I am."
If you take away the yoke from your midst,
the pointing of the finger, and speaking wickedness,
if you pour yourself out for the hungry
and satisfy the desire of the afflicted,
then shall your light rise in the darkness
and your gloom be as the noonday.