Othello includes an embroidered handkerchief that becomes central to the plot in the second act. Performances, such as 1995's film version, include the handkerchief earlier in the film as a part of Desdemona's costume. Shakespeare's plays, as written, do not include prop or costuming instructions, so the necessity to include specific props such as this must be inferred by the director.
How about guns in Chekhov's own plays? In The Seagull, Konstantin Treplyev kills a seagull and brings his rifle on stage. The trope is seemingly subverted when he attempts to use it to suicide and is not successful, but at the end of the play manages to succeed. It is, however, averted in Uncle Vanya, where Uncle Vanya attempts to kill the professor but no clues are given beforehand, and a gun is seen in The Cherry Orchard, but never fired.
Used to great effect in Eugene O'Neill's one act play The Emperor Jones (1920). In the first scene, the eponymous self-proclaimed Emperor explains to another character how he managed to convince all of his subjects that only bullets made of pure silver could hurt him. To demonstrate his arrogance and overconfidence, he pulls out his gun and shows the other character an actual Silver Bullet he commissioned himself which he keeps inside the gun's bullet chamber at all times as a final resort in case the vengeful natives finally catch up with him. Naturally, the silver bullet is used towards the end of the play, but in an ironic twist, the actual bullet itself is used to 'kill' a terrifying hallucination dredged up by the Emperor's own mind. In the play's final scene, the report of the gun has given away his position to the vengeful natives, who, upon locating their hated despot in the middle of a dark jungle, riddle the Emperor full of homemade silver bullets.
Used in Beaumarchais's The Marriage of Figaro: Marcellina makes a throwaway comment in Act I regarding her long-lost son—who is naturally revealed later to be Figaro, conveniently removing his obligation to marry her.
An actual gun belonging to Joe Keller is introduced in the first act of Arthur Miller's All My Sons and reappears in the last scene when Joe shoots himself.
Cyrano de Bergerac: Cyrano’s slight cut at Act II Scene III would be important at Act II Scene VI.
Loves Labours Lost: Berowne mentions in Act I scene 1 that the Princess's father is "decrepit, sick, and bedrid." This never comes up again until five acts of witty banter later, when the father dies, so the ladies have to leave and go into mourning and none of the couples can get married.
In Miss Saigon, Kim shoots her cousin Thuy when he tries to kill her mixed-race son. A flashback later in the show reveals that Chris gave her the gun for protection. This might technically invert this trope, but it's played straight later when Kim uses the same gun to shoot herself in order to force Chris and his wife to take the boy to America with them.
He gets the SuperSoaker Fire Extinguisher, thus fulfilling the Chekhovian unities, to wit: "If you bring on a SuperSoaker Fire Extinguisher, it must eventually go off."
In Oh Dad, Poor Dad, Mamma's Hung You in the Closet and I'm Feelin' So Sad, there is a fire axe present from the beginning of the first act. Jonathan breaks the glass and uses it in the third act.
In Amaluna, Miranda is given a crystal/stone by the Moon Goddess at the beginning of the latter's aerial hoop act. Towards the end of the show, she uses its powers to save Romeo from Cali.
Pokémon Live! has MechaMew2 learning Selfdestruct at the beginning; it uses this in the finale to blow itself up and take down Team Rocket.
In The Wild Duck great attention is paid to the Ekdal's family gun or "That there gun" as Hjelmer Ekdal's wife puts it. In the final act of the play, Ekdal's young daughter Hedvig kills herself with it.
In Pokémon: The Mew-sical, this is parodied. Professor Oak gives Ash a turkey baster, saying he'll know when the time is right to use it. He attempts to use it to cut his bonds, but drops it.