The irony of having to let go of someone you love is that it takes a lot of love to be able let go of someone. This is true in any relationship, whether romantic or familial. When there is a divorce or other kind of breakup, it is necessary to go through a grieving process, the same as if someone had died. This grieving process is necessary no matter how congenial or nasty the breakup.
Problems occur when you avoid the grieving process and try to cling to something that has died. At the very least, the inability to let go will keep you stuck in the past and you won't really be living your life. It is natural to grieve for a period and just as natural to eventually let go.
At it's worst, the inability to let go can turn into some form of stalking, an obsession for what is lost and a desire to know everything that is happening to your former partner. To say that this is unhealthy is an understatement.
Finally, one of the most common ways in which this challenge presents itself is in parents letting go of their children. This is one of the most difficult challenges parents have to deal with emotionally. It is also one that is probably often not successfully dealt with. The desire for parents to hold onto their children probably causes as much damage in the population as any other form of bad parenting, not including actual abuse.
Letting go of someone you love is a way of truly loving them. Ultimately, it is also a way of loving yourself.