Your question wasn't very clear. Assuming the logo you received in .ai is indeed in vector format and not an embedded image, the background should remain transparent when opened in Photoshop. When exporting to png, the transparency option should be clicked by default.
If the designer added a literal white background in .ai you must have Illustrator or other vector software that can open .ai to remove the white background.
If on the other hand, you only want the stroke (outline) of your logo and have the fill be transparent, again you'll need a vector software to make that change. Ideally you don't do it in PS.
My best suggestion is that you contact the designer and ask them to produce png (or .tiff) of the logo with transparent background.