Hi Damon ,
Thank you for contacting Syncfusion Support.
We have checked your query
"I'd like to change the Icon to On when tab Item is hovered over and On when clicked" and we have prepared the sample to meet your requirement. In this sample we have added two images to SfTabItem header and update these image visibility based on mouse enter or exited on SfTabItem. Please download the sample from the below link.
Sample:
SfTabControl_IconShowRegards,
Marimuthu S.