I do not know the source of this rite, nor do I claim any rights to it. If you know the source of this, I’d be glad to add the proper recognition. This is the rite we use for the burning of the palms, the ashes of which are used for the Imposition of Ashes on Ash Wednesday:

Rite for the Burning of Palms – Shrove Tuesday

P: In the name of the Father and of the + Son, and of the Holy Ghost.
R/: Amen.
P: Let us pray.
God of tender mercy, Thou didst fashion us from the dust of the earth, and dost bid us to follow Thy Gospel call in the company of Thy Holy Church. Look Thou upon us with kindness as we prepare the ashes which mark the beginning of our Lenten journey.  Do Thou grant that we who make the desert pilgrimage might receive by Thy grace a renewed passion for Thy Holy Word and Sacraments.  Grant also that our aims may serve well Your people.  Bring us peace, we pray Thee, according to Thy will.  Bless, as Thou hast promised, that by Word and prayer all our needs be sanctified, and that at last we might come to the Celestial Table which Thou hast prepared in the place where Thou reignest with Christ and the Holy Ghost, ever One God, world without end.
R/: Amen.

Scripture Reading + Isaiah 58: 5-11
P: A reading from the Book of the prophet Isaiah.
“Is it such a fast that I have chosen? a day for a man to afflict his soul? is it to bow down his head as a bulrush, and to spread sackcloth and ashes under him? wilt thou call this a fast, and an acceptable day to the LORD? Is not this the fast that I have chosen? to loose the bands of wickedness, to undo the heavy burdens, and to let the oppressed go free, and that ye break every yoke? Is it not to deal thy bread to the hungry, and that thou bring the poor that are cast out to thy house? when thou seest the naked, that thou cover him; and that thou hide not thyself from thine own flesh? Then shall thy light break forth as the morning, and thine health shall spring forth speedily: and thy righteousness shall go before thee; the glory of the LORD shall be thy rereward. Then shalt thou call, and the LORD shall answer; thou shalt cry, and he shall say, Here I am. If thou take away from the midst of thee the yoke, the putting forth of the finger, and speaking vanity; And if thou draw out thy soul to the hungry, and satisfy the afflicted soul; then shall thy light rise in obscurity, and thy darkness be as the noonday: And the LORD shall guide thee continually, and satisfy thy soul in drought, and make fat thy bones: and thou shalt be like a watered garden, and like a spring of water, whose waters fail not.”

P: The Word of the Lord.
R/: Thanks be to God.

Psalm and Burning of Palms
Psalm 51 is chanted or spoken antiphonally and responsively as the palms are burned. The Psalm may be chanted to Gregorian Mode IV . The verses in bold face are said/chanted by the congregation:

Have mercy upon me, O God, according to Thy / lovingkindness: +

according unto the multitude of Thy tender mercies blot / out my transgressions.

Wash me throughly from / mine iniquity, +

and cleanse me / from- my sin-.

For I acknowledge / my transgressions: +

and my sin is / ever before me.

Against Thee, Thee only, have I sinned, and done this / evil in Thy sight: +

that Thou mightest be justified when Thou speakest, and be / clear when Thou judgest.

Behold, I was shapen / in iniquity; +

and in sin did my / mother conceive me.

Behold, Thou desirest truth/  in the inward parts: +

and in the hidden part Thou shalt make / me to know wisdom.

Purge me with hyssop, and / I shall be clean: +

wash me, and I shall be / whiter than snow-.

Make me to hear / joy and gladness; +

that the bones which Thou hast brok-/en may rejoice-.

Hide Thy / face from my sins +

and blot out / all mine iniquities.

Create in me a / clean heart, O God; +

and renew a right / spirit within me.

Cast me not away / from Thy presence; +

and take not Thy Ho-/ly Spirit from me.

Restore unto me the joy of / Thy salvation; +

and uphold me / with Thy free spirit.

Then will I teach trans-/gressors Thy ways; +

and sinners shall be conver-/ted unto Thee-.

Glory be to the Father, and to the Son, and / to the Holy Ghost, +

As it was in the beginning, is now, and ever shall be, / world without end. Amen.

Have mercy upon me, O God, according to Thy / lovingkindness: +

according unto the multitude of thy tender mercies blot / out my transgressions.

When the palms are completely burned, the ashes are sieved or ground to a fine powder. Then the following blessing is given:

 Blessing
P: Lord, bless + we pray Thee these ashes, by which we show that we are dust. Do Thou pardon our sins, and keep us faithful to the discipline of Lent, for Thou  desirest not the death of the  sinner, but rather that we should live with the risen Christ, Who reigneth with Thee and the Holy Ghost, ever one God, world without end.
R/: Amen.