The way I would go about doing this would be to have a counter which starts at 1.
Then in your button click event you would check what the counter was if its 1 then you would put the current selecteditem text into textbox1, if the counter was 2 then into textbox2 and I think you can guess where if the counter is 3 
So your code might look similar to this
Code:
Public Class Form1
Private counter As Integer
Private Sub Button1_Click(ByVal sender As System.Object, ByVal e As System.EventArgs) Handles Button1.Click
If ListView1.SelectedItems.Count = 1 Then
Select Case counter
Case 1
TextBox1.Text = ListView1.SelectedItems(0).Text
Case 2
TextBox2.Text = ListView1.SelectedItems(0).Text
Case 3
TextBox3.Text = ListView1.SelectedItems(0).Text
End Select
counter += 1
'This bit of code is so that you loop round and start at 1 again
If counter > 3 Then
counter = 1
End If
Else
MsgBox("You must select an item in order to continue")
End If
End Sub
Private Sub Form1_Load(ByVal sender As System.Object, ByVal e As System.EventArgs) Handles MyBase.Load
counter = 1
End Sub
End Class
I've gone with the idea that you probably only want one thing selected at once, so you would probably want the listviews MultiSelect property set to false for this.
Anyway I hope that helps you
Satal