Everyone loves Desdemona although Iago manipulates others into thinking she is something less than chaste.
The problems with the handkerchief start when Desdemona, dutifully attempts to make Othello feel better "Let me bind it.." (III.iii.290) at which point the handkerchief is dropped. Unfortunately, Desdemona does not notice that she has mislaid such a precious item but this will ultimately contribute to her demise. Emilia picks it up as she knows of its importance and without having more time to consider her position she gives it to Iago, unwittingly becoming complicit in his evil and destructive plan. Later this will become the "ocular proof" of her suspected infidelity and will seal Desdemona's doom and Othello's ruin.
Othello gave the handkerchief to Desdemona- his first gift to her. It had belonged to his mother and was a cherished item. For Desdemona to lose it so carelessly and without what Othello considers to be true regard for its significance when she discovers that it is lost causes her to
become entangled once again with Othello’s monstrous jealousy.
Desdemona does not expect Othello to be "put to ill thinking"(III.ii.26) by the loss of the handkerchief. Having been spurred on by Iago's claims, Othello tests Desdemona himself, knowing she does not have the handkerchief. He explains how it must never be lost and that
if she (Othello's mother) lost it,/Or made a gift of it, my father's eye/ Should hold her loathely..."(III.iv.60-62)
This foreshadows Desdemona's own death as "To lose't or give't away were such perdition/ As nothing else could match."(III.iv.67-68). Othello, the Moor, is susceptible to a belief in charms and as there's "magic in the web of it" he is easy prey which makes Iago's
manipulation of Othello more devious because he plays on Othello’s vulnerability
The handkerchief is therefore instrumental in Desdemon's death. She does however try to explain to Othello but by this stage he is so enraged and not thinking rationally, believing that the only way to save her is to kill her as "so sweet was ne'er so fatal."(V.ii.20)