What happens when a Protestant doesn't tithe? I heard in another question that some Protestants have to tithe. That is, they must give one-tenth of their salary, or more broadly, a specific amount or percentage. That seems a bit strange to me, since I've never had to tithe. I'm not saying it's a bad practice (after all, there was a time at which God ordained it). I just don't really have any proper conception of how it works among Protestants.

Are tithing Protestants supposed to submit proof of their salary? What happens if they don't meet the mark? Are they said to sin, and if so, how serious is the sin? Are they barred from entering their parish? Does the pastor frown at them or is some other form of social pressure applied? Are they asked to come in for a meeting to explain their financial circumstances? How does it work?
No Protestant churches that I know of ever require members to submit proof that they are tithing (though there certainly may be some).  Tithing is generally considered to be something that the members ought to do, but is between them and God.  

This may flow out of the doctrine of the Priesthood of All Believers.  Each person is directly accountable to God for his life and actions.  

In 2 Corinthians 9:7 (as David Morton mentioned in his answer), the Scriptures teach us how to give:


  Each one must give as he has decided in his heart, not reluctantly or under compulsion, for God loves a cheerful giver  2 Corinthians 9:7


A key part of this is the phrase not reluctantly or under compulsion.  If someone is put under compulsion to give, but his heart is not in it, his giving is not considered to be a righteous act or a good deed in God's eyes.