How to continue running code after an exception is thrown?

Although On Error Resume Next is still available in VB.NET, it is mutually exclusive to the preferred method of structured exception handling.

Instead, I would recommend the use of the Finally clause of a Try..Catch..Finally block to ensure Line 5 and Line 6 get executed even if Line 4 (or any preceding Line) throws.

Try
  line 1
  line 2
  line 3
  line 4
Catch ex as Exception
   log(ex.tostring)
Finally
  line 5
  line 6  
End Try

If you're doing something that you know how to recover from or that isn't vital, you're supposed to wrap just that line in the try/catch with a specific catch. e.g.

Try 
  line 1
  line 2
  line 3
  Try
     line 4 ' (here the exception is throw and jumps to the catch)
  Catch iox as IOException ' or whatever type is being thrown
     'log it
  End Try
  line 5  ' <-- I would like the program to continue its execution after logging the error
  line 6  

Catch ex as Exception
   log(ex.tostring)
End Try

Use 'Continue For'

Not good practice everywhere, but useful in some circumstances, e.g. find a file while handling denied access to certain directories:

    Dim dir As New DirectoryInfo("C:\")
    Dim strSearch As String = ("boot.ini")

    For Each SubDir As DirectoryInfo In dir.GetDirectories
        Try
            For Each File As FileInfo In SubDir.GetFiles
                Console.WriteLine("Sub Directory: {0}", SubDir.Name)
                If File.Name = strSearch Then
                    Console.Write(File.FullName)
                End If
            Next
        Catch ex As Exception
            Console.WriteLine(ex.Message)
            Continue For
        End Try
    Next