The Vikings burying the chest containing the mask on an American beach.
Peggy's death after betraying Stanley, where Dorian (who has the Mask on) hurls Peggy into a printing press, gets mashed inside of it, and comes out as a bloodied newspaper reporting her death. Because of that, Peggy just disappeared from the plot after getting her money and didn't get what she deserved, though, had they kept that scene in, it probably would have given the movie an "R" rating, and there would be a Continuity Snarl in the animated series, where Peggy is alive and well, and still remembers the time she betrayed Stanley.
An extended scene concerning The Mask's confrontation with a street gang.
An early workprint of the movie can be found online which has more extended/deleted scenes not available on the DVD. One of which is a scene that shows Stanley getting dropped off by his apartment after finding The Mask in the Edge City river, and is immediately confronted by the same street gang that he would see again shortly as The Mask. The gang's leader asks Stanley for the time so he can steal his watch, foreshadowing how Stanley would soon get payback on him as The Mask when asked the same question.
With that in mind, remember this exchange?
Mrs. Peenman: "Ipkiss! Do you have any idea what time it is?!"
Ipkiss: "Actually, no."
Executive Meddling: Very surprisingly inverted, at least if one listens to the DVD commentaries/behind the scenes material. Chuck Russell — director of A Nightmare on Elm Street 3: Dream Warriors stated that New Line was interested in developing "The Mask" along the lines of the comic books; this would have become, according to Russell, "the next Freddy Krueger". But, he noted, that draft after draft of the more comics-faithful adaptation simply did not work. So, he fought valiantly to make "The Mask" as a comedy; the Cuban Pete sequence is one of the few elements that survived until the final cut.
There were plans for an actual sequel (not, Son of the Mask, mind you). One idea was that Lt. Kellaway finds the mask, dons it, and becomes the next villain. Furthermore, Nintendo Power actually had a contest that whoever won could appear in the film as an extra. They Lampshaded this in their final issue.