There are two types of people who threaten to leave a church, and how a church responds depends upon which category they would fall into. As a pastor the ones I hated to see leave were the ones who simply were not comfortable with the changes that were occurring in the church. These were good people who did not want to create problems, but the new directions the church was taking were not what they wanted in a church. It affected their ability to worship and to serve, and as painful as it is to lose such people it is probably best for them and the church for them to leave.
I was working with a church one time that was trying to make a decision between two options. At a leadership meeting one night one of the leaders said that if the church chose one option he and his family would leave. He wasn't angry, and he didn't say it as a threat. He just wanted the others to know that his family would not be comfortable with that choice. I immediately said to the group, "And that's OK. If he and his family chooses to leave that is their choice to make and should not influence the decision you make." In the end, the church did make the choice that he was not comfortable with, and he and his family left that church. Within a few weeks they settled into another nearby congregation where they continue to be very active members and hold leadership positions. You hate to lose such people, but if God is leading one way and they felt led another direction you have to respect that and make the decision that God is leading the entire congregation to make.
The second group consists of the controllers. Any time they don't get their way they stomp and snort telling everyone they are going to leave the church. These are people who believe the church exists for them and their families, and everyone else there is expected to bow down to their wishes. They may or may not be major givers in the church (sometimes you might be surprised to find out that the ones who demand the most give the least). They will announce to anyone who will listen that they represent a large group within the church, and if they leave that whole group will leave. Sometimes that happens, but in my experience I've learned that they usually represent their family who were conditioned years ago to not cross them and few others outside their immediate family. Even if a large group does follow them out of the church, are these people with whom you would expect to build a ministry? If they are following a controller that means they are not following God.
I would never lose a minute's sleep when these people leave. I would wish them well and then turn back to the task God had given me. While I would never invite people to a party to celebrate that these people had left, I might be tempted to thank God for a blessed reduction in our church. Sometimes a church needs to grow a little smaller before it can grow larger.
A church will never be any healthier than its members and its leaders. If your church consists primarily of controllers and other dysfunctional people it will never be healthy until these people are gone, and until it has a measure of health it will not be able to do much for the Kingdom of God. For more on healthy churches and how to deal with controllers you may want to read my book The Healthy Small Church: Diagnosis and Treatment for the Big Issues.