What an interesting question! In this example, to tuck something is to put it into a (small) space. You can substitute the word "put". • She tucked a coin into the boy's palm. • I tucked a napkin under his sandwich. It's not necessarily a metaphor because she really did put something under her pillow and either she WAS ashamed or she should have been! What was it? A note from her lover? A photo? Everything up to that line tells us that the person singing the song has reason to doubt her faithfulness, so she was probably up to something!!!

"I saw something tucked in shame underneath your pillow" - in this context, it means something is hidden under the pillow, because the person hiding it is ashamed of what they have done or what they have hidden.