A ValueConverter would work. Another option would be to use a DataTrigger in the style of ListBoxItem. Maybe something like this:
<Style x:Name="MinDateTimeListBoxStyle" TargetType="ListBoxItem">
<Style.Triggers>
<Setter Property="Background" Value="Gray" />
<DataTrigger Binding="{Binding Path=Done}"
Value="{x:Static sys:DateTime.MinValue}">
<Setter Property="Background" Value="White" />
</DataTrigger>
</Style.Triggers>
</Style>
This will set the background to Gray when the value of Done isn't DateTime.MinValue. I don't think there is a way to do a not equals comparison in a trigger, so it sets the background to Gray by default, and only changing it back to white if Done hasn't changed yet. It would probably be better to use the correct color for the background instead of white (maybe get the value of the parent's background?), but this should give you something to start with.
Update: To apply this style to the items of only certain ListBoxes, give the style a name and set the ItemContainerStyle as appropriate:
<ListBox x:Name="StyledListBox"
ItemContainerStyle="{StaticResource MinDateTimeListBoxStyle}" />
<ListBox x:Name="NormalListBox" />