When I do the following (for example):

(put-text-property (point) (1+ (point)) 'display (propertize "." 'face
'header-line))

The character at point becomes a dot with a gray background. If I then
enable hl-line-mode, the line turns green, as expected. However, the dot
still has a gray background; it does not turn green with the rest of the
line.

It appears that hl-line-mode uses an overlay to highlight the line. From
the overlay documentation, "Currently, all overlays take priority over text
properties." But it seems like in this case, the display text property is
taking priority over the the overlay. Since I'm embedding the face into the
display string, I expect it to take priority over any faces that are
applied to the text, but not over any overlays which affect it.