How to trigger DataTemplateSelector when property changes?

As you requested an example with datatriggers in the comments, here you are:

A FrameworkElement can only have EventTriggers, therefore you get the error Message Triggers collection members must be of type EventTrigger

And also don’t use a ContentPresenter directly, it is meant to be used inside a ControlTemplate. Better use a ContentControl when you want to have dynamic content.
See What’s the difference between ContentControl and ContentPresenter?

And finally here’s a suggestion to your DataTrigger issue. I have put it inside a style for reusability ….

XAML :

<Window x:Class="WpfApplication88.MainWindow"
        xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"
        xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"
        Title="MainWindow" Height="350" Width="525">
  <Window.Resources>

    <DataTemplate x:Key="requiredTemplate">
      <TextBlock Text="requiredTemplate"></TextBlock>
      <!--your stuff here-->
    </DataTemplate>

    <DataTemplate x:Key="completedTemplate">
      <TextBlock Text="CompletedTemplate"></TextBlock>
      <!--your stuff here-->
    </DataTemplate>

    <Style x:Key="selectableContentStyle" TargetType="{x:Type ContentControl}">
      <Style.Triggers>
        <DataTrigger Binding="{Binding Path=CurrentStatus}" Value="Required">
          <Setter Property="ContentTemplate" Value="{StaticResource requiredTemplate}" />
        </DataTrigger>
        <DataTrigger Binding="{Binding Path=CurrentStatus}" Value="Completed">
          <Setter Property="ContentTemplate" Value="{StaticResource completedTemplate}" />
        </DataTrigger>
        <!--  your other Status' here -->
      </Style.Triggers>
    </Style>

  </Window.Resources>

  <Grid>
    <ContentControl Width="100" Height="100" Style="{StaticResource selectableContentStyle}"/>
  </Grid>

</Window>