In approaching this question from another angle, one could argue that Iago is also a tragic character.  Throughout the play, Iago is jealous of Cassio's new position and feels that he should have been given the title by Othello.  It is in part jealousy that drives Iago to manipulate events against Othello so that he can take revenge upon Othello for favoring Cassio.  Before his scheming, Iago appears to have been a highly favored officer and he is well-liked and respected by his peers.  He is blinded by his jealousy, and in the end, all is revealed and Iago must suffer the consequences.  Therefore, Iago fits the model of a tragic character--one who began honorably and is overtaken by a tragic flaw.

It has been questioned whether Othello deserves the title of a tragic character as he is not recognised as of noble birth by the society in which he exists. Othello is a well-respected and successful military figure in Venice, but he is not a member of the royal family.

 Herein may lie some of the jealousy; from Othello to the position of those around him, who were born into their ranks rather than making themselves 'noble' as he has. This contrasts with the jealousy of 'Iago', tortured in the belief that Othello has denied him his rightful promotion to lieutenant, and may have slept with his wife, Emilia. Othello’s jealousy towards Cassio and the believed affair with Desdemona may be rooted in Othello’s insecurity in a society where he is still regarded as an outsider by some.

Love comes in such a wide range of forms in Shakespeare’s plays – courtly, familial, sexual, platonic, filial, to name but a few. You would need to identify the characters you wish this theme to apply to in order to develop a refined answer.

In terms of a tragic character being a person who falls from grace or high status into a downward spiral that leads to his doom, then we must surely be talking about Othello himself here. Here's a man who has clawed his way up from the lowest of the low (in those times it was sometimes possible for a man to work or buy his way out of slavery/tied level through an successful rise in  the army ranks) to a much more highly-praised and highly-paid general in the army. At this level he would have been hobnobbing with the elite (politicians,strategists,maybe even royalty through praise.) He has friends and loyal supporters. He is alone though, in terms of romantic emotional suppport and love. Then he meets Desdemona and an erstwhile colleague has reason to find hate and jealousy... Othello's fall is swift and catastrophic.