Disable Text Selection in TextBox Style

Well, one quick way to disallow text from being highlighted is to add to your styles toggle of IsEnabled. Otherwise, you might want to override a TextBox's Template, with a StackPanel, that toggles the TextBox and a TexBlock - super easy to do, and ton of samples all over, even here on stack overlow. Here's a your modified style with IsEnabled..

<Style x:Key="TextBlockStyleForTextBox" TargetType="{x:Type TextBox}" BasedOn="{StaticResource {x:Type TextBox}}">
        <Setter Property="Background" Value="Transparent" />
        <Setter Property="BorderThickness" Value="0" />
        <Setter Property="IsReadOnly" Value="True" />
        <Setter Property="IsEnabled" Value="False"/>
    </Style>

    <Style x:Key="RenamingTextBox" TargetType="{x:Type TextBox}" BasedOn="{StaticResource TextBlockStyleForTextBox}">
        <Setter Property="Cursor" Value="Arrow"/>
        <Setter Property="Padding" Value="0" />
        <Setter Property="Margin" Value="0" />
        <Style.Triggers>
            <DataTrigger Binding="{Binding IsRenaming}" Value="true">
                <DataTrigger.Setters>
                    <Setter Property="TextBox.IsReadOnly" Value="False" />
                    <Setter Property="IsEnabled" Value="True"/>
                    <Setter Property="Cursor" Value="IBeam" />
                    <Setter Property="Background" >
                        <Setter.Value>
                            <SolidColorBrush Color="{DynamicResource WhiteColor}"/>
                        </Setter.Value>
                    </Setter>
                    <Setter Property="BorderThickness" Value="1" />
                    <Setter Property="Padding" Value="2" />
                </DataTrigger.Setters>
            </DataTrigger>
        </Style.Triggers>
    </Style>

Your styles are working just fine, as I just tested them! 3 possible problems:

1. You are not binding the TextBox correctly
2. Your are not assigning your style

3. IsRemaining isn't raising property changed

   ---

   here's my test

   <Window x:Class="Sample.MainWindow"
   xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"
   xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"
   Height="500" Width="600" 
   x:Name="wnd">
      <Window.Resources>
          ...I put your styles in there..
      </Windows.Resources>
   <StackPanel>
       <TextBox DataContext="{Binding ViewModel, ElementName=wnd}" Style="{StaticResource RenamingTextBox}" >tata</TextBox>
       <Button Command="{Binding ViewModel.SomeCommand, ElementName=wnd}">change read only</Button>
   </StackPanel>
   

code behind (.xaml.cs)

public partial class MainWindow : Window
{
    public ViewModel ViewModel { get; set; }

    public MainWindow()
    {
        ViewModel = new ViewModel();
        InitializeComponent();          
    }
    }

public class ViewModel : INotifyPropertyChanged
{
    private DelegateCommand _someCmd;
    private bool _isRenaming;

    public DelegateCommand SomeCommand
    {
        get
        {
            return _someCmd ?? (_someCmd = new DelegateCommand(() =>
                                                                {
                                                                    IsRenaming = true;
                                                                }));
        }
    }

    public bool IsRenaming
    {
        get { return _isRenaming; }
        set
        {
            _isRenaming = value;
            RaisePropertyChanged("IsRenaming");
        }
    }
      }