When a data provider is used, the designer does not show the RadRichTextBox content, that is shown with no problem at run time.

Workaround: The issue is with finding the format provider used by the data provider. This issue appears at design time only. The workaround is to manually set the format provider of the data provider:
        <telerik:HtmlDataProvider
            x:Name="htmlDataProvider"
            RichTextBox="{Binding ElementName=radRichTextBox, Mode=OneWay}"
            Html="&lt;h1&gt;Header3&lt;/h1&gt;">
            <telerik:HtmlDataProvider.FormatProvider>
                <telerik:HtmlFormatProvider />
            </telerik:HtmlDataProvider.FormatProvider>
        </telerik:HtmlDataProvider>