Did you know you can remove the background for a PNG image?
Use the Preview app to create a transparent background
Alternatives to explore
Apart from PNG and JPEG, there are other approaches to displaying logos that may fit better for your store.
Use a GIF image
For some Pixel Union themes, such as Superstore, GIF format images can be displayed in the header. These GIF images can animate and may work to the advantage of certain brands. Some key disadvantages of GIF images are their size and scalability. In other words, they take longer to load the image file and don't adapt to smaller screens as well as the recommended PNG or JPEG.
Modify the theme files to display an SVG
If using an SVG for your logo is important to your store, you'll need to hire a 3rd-party development service like Storetasker to alter your theme files for this purpose. Pixel Union themes are incompatible with this graphic format unless modified otherwise.