Originally the film started with the arrival of Duncan on Arrakis.

From the "Art and Soul of Dune" book we have a picture of Jessica training Paul on Caladan.

Paul says goodbye to his horse on Caladan, and gifts the horse to the stable boy.

In a mediation rooom in the Arrakeen residency, Jessica teaches Paul the Voice, to demostrate she commands Paul to "Drink the flame" of the oil lamp.

Gurney baliset can be seen briefly in the movie next to his bed, however Hans Zimmer did compose a melody using lyrics adapted from the novel and a scene was filmed with actor Josh Brolin "playing" the prop baliset.

According to "Art and Soul of Dune" book our first view of Stilgar in the movie was originally Stilgar walking in the desert when he spots a Heighliner in the sky, bringing the Atreides to Arrakis.

When the Atreides arrive on Arrakis they find a box left by the Baron as a gift. Opening it they find it contains the fingers of many Spice workers.

Looking closely at the monitors in this behind the scenes picture it appears to show Gurney holding a knife to a Harkonnen in the Arrakeen Residency, the "Box of fingers" is still sitting on the table behind them.

This picture appears to show Doctor Yueh giving Paul Atreides a copy of the Orange Catholic Bible.

This picture appears to show Lady Jessica and Doctor Yueh talking, possibly discussing Yueh's wife, Wanna.

According to "Art and Soul of Dune" book this picture is of Liet Kynes visiting Duke Leto in his office in Arrakeen.

"I don’t know whether it’s still in the film or not, but in the script that we had, Paul said to him, ‘It was also what you taught me that made me triumphant over that hunter-seeker,'” Henderson says. “So that’s the only thing that helped [Thufir] get off the hook, but [his] resignation was a sincere desire to say, ‘Get someone else because I care that much.'"

Hollywood Reporter

The script contained a short scene of the Atreides searching for the Hunter Seeker operator. Territory Studio created animation for it, but it was ultimately unused.

Set on the balcony of the Arrakeen Residency, Paul expresses his concerns to Leto that his life is already set by destiny.
Picture courtesy of Zsuzsanna Virág.

INT. ARRAKEEN RESIDENCY - WAR ROOM - END OF DAY
Duke Leto stands in front of Duncan Idaho. Leto communicates with precise hand signals using THE ATREIDES COMBAT SIGN LANGUAGE. We can read the translation with subtitles.
Leave at dawn. Give this message to the FREMEN.
The Duke gives him a tiny object.
Trust no one.
Duncan bows and leaves. Leto stays alone, lost in thoughts.

In an interview with Variety, actor Stellan Skarsgård confirmed that there was a scene with the Baron and Jessica that was cut from the movie.

LETO'S BEDROOM - NIGHT
Jessica lies on the floor in her nightgown. Ankles tied,
wrists bound behind her back. Mouth gagged. Her eyes closed.
...

In some Behind The Scenes footage we see Piter drinking what appears to be Sopho Juice.

In some Behind The Scenes footage we see Piter with the Atreides Mentat Thufir Hawat captured behind him. "There was also a scene between myself and Thufir Hawat, played by the great Steven McKinley Henderson, which unfortunately didn't make it into the final cut of the film."

When the Baron learns of the Atreides attack on his Spice stockpile (another deleted scene) he rises out of his healing bath.

The Baron floats in the air: naked but for his suspensor harness. Entirely covered in glistening black fluid, even his head and face. He is breathing like a bellows, chest heaving.

In some Behind The Scenes footage we see Chani imparting Fremen wisdom to Paul. "Do not brush against a bush lest you leave a thread to show the way, where we go no one must follow."