Favicons are stored in the Temporary Internet Files folder. If IE is set to clear this folder upon closing, then the icon will be lost. After you create your shortcut on your Desktop and before closing IE, do this:
Open your Temporary Internet Files folder (Location is shown in IE > Tools > Internet Options > General tab > section Temp Internet Files > Settings). Search this folder for *.ico. You will see your icon with the name Favicon. Copy this file and paste it into a folder of your choice, like C:\My Documents\MyFavicons. Rename the icon to be meaningful to you. The file type must be "ico"
To install the icon permanently, right-click the link of the shortcut you saved on the Desktop and select Properties. Select Change Icon. Select Browse and go to the MyFavicons folder and select the icon you placed there. Click Apply and OK out. Now you can move the shortcut anywhere and you will retain the icon.