There's no hard-and-fast rule about this; it really varies on the circumstances. BUT, over the years, I've noticed something about healthy breakups: the couple gets over each other and goes on with life (usually after a time of being all emo and listening to sad music). The really unhealthy breakups are the ones where the guy or girl keeps obsessing over the other person or the relationship.
The best way to heal from a breakup and get over a person (in a healthy way) is to spend some time apart. In fact, I usually recommend that a couple not talk, text, or spend any time together for at least a month for each year they dated. Once that "cooling off" period has passed, then the couple can talk about whether they want to remain friends and what that could look like.
I know this might sound a bit extreme, but there are three very good reasons not to remain friends with your ex right after a breakup.
1) Talking and spending time with your ex makes it harder to move on.
The toughest but healthiest thing a person needs to do after a breakup is re-learn how to live life without the other person. You have to unlearn old habits, schedules, and emotions, and then replace them with new ones. For example, you might have to set up a new bedtime routine ("we used to talk every night") or figure out a new way to deal with emotions ("I would usually call them when I was lonely or bored").
When you stay friends with your ex right after a breakup, you are VERY likely to fall into old habits and not work at developing new ones. It may feel good in the moment, but it's hurting your long-term growth and healing.
2) It's a REALLY bad idea to turn to your ex to reduce the pain of a breakup.
ave you ever wondered why so many couples get back together shortly after a "breakup"? It's because all of us turn to whatever is enjoyable, comfortable, and familiar during times of emotional or physical stress. (Why do you think we turn to our favorite comfort foods when we're sad?)
For a dating couple, that thing we usually turn to for comfort is the other person. So, even though a relationship has ended, our natural reaction will still be to turn to our ex in order to ease the pain. Simply put: if you're hurting because of your ex, it's a really bad idea to go to your ex to make those bad feelings go away. You're not healing the cause of the pain; you're just making it worse.
3) We usually lie to ourselves about our true motives for remaining friends.
Most Christians will have very noble, biblical-sounding reasons for remaining friends with their ex right after a breakup: "they're my brother/sister in Christ", "I've forgiven them", "we've reconciled". But most of the time they're doing it because life without their gf/bf is too painful, and they're still trying to get something from their ex that they got while dating (like comfort, attention, affection, validation, etc.) You have to be brutally honest with yourself why you want to remain friends with them.
So, here's my answer simplified into one sentence: It is okay to stay friends with your ex, but only after a period of separation so that you are not trying to re-live the same emotions of the dating relationship but calling it a "friendship."