pmason,
The logo image becomes a web part property when the web part is placed on a web part page and must be removed as you describe in your question on a web part by web part basis. However, to prevent the logo from becoming a web part property in the future, you may edit the web part XML schema file in the feature directory and remove the reference to the logo. You must then perform an IIS reset. This will prevent the web part from taking the logo link reference as a property when it is placed on a web part page...preventing our logo from appearing in the title.
You will need to locate the file named bWareTech.WhatsNew.webpart located in the web parts feature directory. If your SharePoint installation was installed in the default location, the feature directory for the web part will be C:\Program Files\Common Files\Microsoft Shared\Web Server Extensions\12\TEMPLATE\FEATURES\bWareTech.WhatsNew\WhatsNew. Open the file for editing in Notepad or an XML editor and remove the following line (element).
<property name="TitleIconImageUrl" type="string">/_layouts/images/bWareTech/bWareTech16.png</property>
Removing this element will allow the web part to be placed on a web part page without a value in the Title Icon Image URL ToolPart property. Please note that you will have to perform this operation on every SharePoint front end web server that has the web part Solution installed. Also note that if/when you install a future update, your changes will be overridden and this operation will need to be performed again.