Increased pressure and ticking clocks, relationship friction, losing an advantage, ego-related conflict

Dating someone who isn't exclusive
A love interest's old flame showing up and wanting to resume a relationship
An ex trying to win back the love interest
A rival who wants the same love interest
Being 'just friends' yet wanting more (when competition shows up)
A rival seeking ways to hurt the character, including stealing their love interest

Having to find ways to one-up the competition
Having to go to a special event solo because they acted too slowly
Being distracted (which interferes with work, school, and other responsibilities)
Having to work twice as hard to be noticed
Being teased or pitied by friends because of the situation
Time lost to worry and anxiety
Jealousy creeping into the relationship, causing arguments
Discomfort at putting themselves out there (if they have not made their feelings known yet to the love interest)

Jealousy that gets out of  hand and causes a breakup
Getting caught spying on the love interest (while they are out with the competition)
Becoming so obsessive it drives the love interest toward the competition
Demanding the love interest make a choice and they choose the competition
Trying to "buy" affection and it turns the love interest off
Pretending to be interested in someone else and it backfiring
Getting into an altercation with a rival and the love interest walks away from both
Giving up and then living with regret
Breaking up and developing a new emotional wound: unrequited love

AGITATION, ANGER, ANGUISH, ANTICIPATION, ANXIETY, BITTERNESS, CONFLICTED, CONFUSION, DEFEAT, DEPRESSED, DESIRE, DESPAIR, DESPERATION, DETERMINATION, DISAPPOINTMENT, DOUBT, EMASCULATION, ENVY, FEAR, GRIEF, GUILT, HOPEFULNESS, HUMILIATION, HURT, INADEQUACY, INSECURITY, INTIMIDATION, JEALOUSY, LONELINESS, LONGING, LOVE, OBSESSION, PARANOIA, POWERLESSNESS, REGRET, RELIEF, RESENTMENT, RESIGNATION, SCHADENFREUDE, SCORN, SELF-LOATHING, SELF-PITY, SHAME, VALIDATION, VALUED, VULNERABILITY, WORRY, WORTHLESSNESS

Insecurity causing neediness (and self-loathing for that neediness)
Anguish over the partner's indecision
Feeling "not good enough" but also angry for being made to feel this way
Anger at the rival but also shame as he (or she) is a good person
Wrestling with trust issues regarding one's love interest
Depression if one is not chosen but also being happy that someone they care about is with the right person
Wanting to share negative information about the rival but knowing the love interest will see it as jealousy
The temptation to cross a moral line to come out ahead

The character themselves, the love interest, rival, third-party individuals who may also have a stake in the outcome (perhaps someone who has feelings for the character but has not made it known to them yet)

CATTY, CHILDISH, CONFRONTATIONAL, CONTROLLING, DISHONEST, FOOLISH, IMPULSIVE, INSECURE, MELODRAMATIC, NEEDY, NOSY, OBSESSIVE, PARANOID, POSSESSIVE, PREJUDICED, PRETENTIOUS, PUSHY, REBELLIOUS, RECKLESS, SUSPICIOUS, TEMPERAMENTAL, WHINY

Esteem and Recognition: Discovering a competitor can create all sorts of problems with the character's self-esteem. Having someone to measure themselves against can shake their confidence, bring out insecurities, and play havoc with their emotions. This in turn might cause them to behave erratically, doing things they might not normally do that only later they will regret, causing embarrassment, humiliation, or even shame.
Love and Belonging: Clearly a competitor showing up can shake the foundations of an existing relationship...depending on the love interest's reaction to this competitor. If they are entertaining the idea of trading your character in for someone else that's going to leave a mark. Or your character loses the love interest to the competitor perhaps because they never let their feelings be known, this will lead to feeling alone and unlovable. 

ABANDONMENT OVER AN UNEXPECTED PREGNANCY, A TOXIC RELATIONSHIP, BEING HUMILIATED BY OTHERS, CAVING TO PEER PRESSURE, GETTING DUMPED, MISPLACED LOYALTY, POOR JUDGMENT LEADING TO UNINTENDED CONSEQUENCES, UNREQUITED LOVE

INNOCENT, INTROVERTED, TRUSTING

A character who has been holding back out of a fear to commit could come to the realization that this was unfair to the other person and make changes moving forward 
The appearance of a rival may force the character to reflect on whether this relationship is worth fighting for or not
A character struggling to show or articulate feelings may finally find the inner strength to push through the mental blocks holding them back
For a character who has a hard time seeing their own strengths, having to put themselves into the love interest's will help them catalog their better qualities, creating a greater sense of self-worth and confidence.