Is there an engine with a dating/relationship system, or do I need a script? I'm looking for something as in Harvest Moon.

Beat me up if I'm wrong, but I'm pretty sure you could event this.

Yup, eventing it is the easiest way. Super easy.

Just make a list of variables called "Joe Relationship" etc. with all the different people. Then, as you talk to them or do things with them that decrease or increase your relationship, change that number accordingly.