I had the same problem using the datasets/tableadapters generated with the designer shipped with System.Data.Sqlite.dll version 1.0.82.0 -- after closing the connection we were unable to read the database file using System.IO.FileStream. I was disposing correctly both connection and tableadapters and I was not using connection pooling.
According to my first searches (for example this and this thread) that seemed a problem in the library itself -- either objects not correctly released and/or pooling issues (which I don't use).
After reading your question I tried to replicate the problem using only SQLiteCommand objects and I found that the problem arises when you don't dispose them. Update 2012-11-27 19:37 UTC: this is further confirmed by this ticket for System.Data.SQLite, in which a developer explains that "all SQLiteCommand and SQLiteDataReader objects associated with the connection [should be] properly disposed".
I then turned back on the generated TableAdapters and I saw that there was no implementation of the Dispose method -- so in fact the created commands were not disposed. I implemented it, taking care of disposing all the commands, and I have got no problem.
Here's the code in C#, hope this helps. Please note that the code is converted from the original in Visual Basic, so expect some conversion errors.
//In Table Adapter    
protected override void Dispose(bool disposing)
{
   base.Dispose(disposing);
    Common.DisposeTableAdapter(disposing, _adapter, _commandCollection);
}
public static class Common
{
    /// <summary>
    /// Disposes a TableAdapter generated by SQLite Designer
    /// </summary>
    /// <param name="disposing"></param>
    /// <param name="adapter"></param>
    /// <param name="commandCollection"></param>
    /// <remarks>You must dispose all the command,
    /// otherwise the file remains locked and cannot be accessed
    /// (for example, for reading or deletion)</remarks>
    public static void DisposeTableAdapter(
        bool disposing,
        System.Data.SQLite.SQLiteDataAdapter adapter,
        IEnumerable<System.Data.SQLite.SQLiteCommand> commandCollection)
    {
        if (disposing) {
            DisposeSQLiteTableAdapter(adapter);
            foreach (object currentCommand_loopVariable in commandCollection)
            {
                currentCommand = currentCommand_loopVariable;
                currentCommand.Dispose();
            }
        }
    }
    public static void DisposeSQLiteTableAdapter(
            System.Data.SQLite.SQLiteDataAdapter adapter)
    {
        if (adapter != null) {
            DisposeSQLiteTableAdapterCommands(adapter);
            adapter.Dispose();
        }
    }
    public static void DisposeSQLiteTableAdapterCommands(
            System.Data.SQLite.SQLiteDataAdapter adapter)
    {
        foreach (object currentCommand_loopVariable in {
            adapter.UpdateCommand,
            adapter.InsertCommand,
            adapter.DeleteCommand,
            adapter.SelectCommand})
        {
            currentCommand = currentCommand_loopVariable;
            if (currentCommand != null) {
                currentCommand.Dispose();
            }
        }
    }
}
Update 2013-07-05 17:36 UTC gorogm's answer highlights two important things:
- according to the changelog on the official site of System.Data.SQLite, starting from version 1.0.84.0 the above code should not be needed, since the library takes care of this. I haven't tested this, but in the worst case you only need this snippet: - //In Table Adapter    
protected override void Dispose(bool disposing)
{
  base.Dispose(disposing);
  this.Adapter.Dispose();
}
 
- about the implementation of the - Disposecall of the- TableAdapter: it is is better to put this in a partial class, so that a dataset regeneration does not affected this code (and any additional code you may need to add).