Transparent Background

Eden,

Based on a different interpretation, it seems that you have paths with white fill as part of the artwork, and that you need to remove those white fills (or paths).

Or, based on another different interpretation, you wish to have the white fills knock out the underlying parts of  the navy blue artwork, in which case you can Group those white paths and select them along with the navy blue parts and apply an Opacity Mask with Clip unticked and Invert Mask ticked (through the Transparency palette), which ought to make the Transparent Grid or any background show through.

If you can show a screenshot with the expanded Layer (Layers palette), it can help helpers.

"when I make the photo transparent or export it as a PNG"

I am afraid you need to export as PNG to keep the transparent parts, which will be turned white with JPEG.

View > Hide Transparency Grid

Unless they are real objects masquerading as a transparency grid.