You will have to update your animation as below i.e give elementname:
<DataTemplate.Triggers>
<DataTrigger Binding="{Binding Removing}" Value="True">
<DataTrigger.EnterActions>
<BeginStoryboard>
<Storyboard>
<DoubleAnimation Duration="0:0:1" From="1.0" To="0.0"
Storyboard.TargetProperty="Opacity" Storyboard.TargetName="ItemBorder" />
</Storyboard>
</BeginStoryboard>
</DataTrigger.EnterActions>
</DataTrigger>
</DataTemplate.Triggers>
OR
Try putting animation directly on your border style like below:
<DataTemplate DataType="{x:Type MyApp:Item}">
<Border x:Name="ItemBorder">
<Label Content="{Binding Path=Name}" />
<Border.Style>
<Style TargetType="Border">
<Style.Triggers>
<DataTrigger Binding="{Binding Path=Removing}" Value="True">
<DataTrigger.EnterActions>
<BeginStoryboard>
<Storyboard>
<DoubleAnimation Duration="0:0:1" From="1.0" To="0.0"
Storyboard.TargetProperty="Opacity" />
</Storyboard>
</BeginStoryboard>
</DataTrigger.EnterActions>
</DataTrigger>
<Style.Triggers>
</Style>
</DataTemplate>