Hi B. Koring,
I'm afraid PrestaShop engine every time would convert your beautiful png/gif logo with transparent background into compressed jpg file with white color behind.
However there is easy to follow workaround to gain effect you need. I'm basing on 1.5.1 but I assume it could work in other versions too (I will mark where are differences between 1.4 and 1.5)
1. Go to back-end panel and upload your logo.
2. Use your favourite ftp/sftp client and open directory:
[ROOT]/img/
in your PrestaShop files.
3. Upload your logo manually. You'll see your file uploaded through back-end panel, i.e. 'logo.jpg'. Upload your 'logo.png' next to existing file.
4. Still with your ftp client go to:
[ROOT]/themes/your_theme_name/
folder and open header.tpl.
5. Find code:
src="{$logo_url}"
in line 77 and change it to:
src="{$img_ps_dir}logo.png?{$img_update_time}"
(update time is optional) [in 1.4, if I remember correctly, your original code would look like: "src="{$img_ps_dir}logo.jpg?{$img_update_time}"']
6. You'll have to repeat these steps every time you would like to change your logo.
BR
Sebastian