I have created a logo using overlapping circles which have a white outline. I have put a colour burn onto the circles. It looks as I would like on illustrator however, when I save it as any kind of file other than ai file or even paste the vector directly into inDesign, an outline of the 'underlayed' circle stroke can be seen where it shouldn't. I have attached an image below which shows a dashed line in the purple area. This is it in inDesign. The version in illustrator doesn't have this dashed line showing through.
Most likely a preview issue. Try switching to a high quality display in Indesign (CTRL+ALT+H).
Also, it is generally recommended you place your logo and any other external files in InDesign, instead of copy-pasting these.
As a sidenote to be 100% safe for logos you should take the time and convert all your outlines to solid shapes, and also remove any opacity settings and instead crop the shapes in your logo so each color is set to normal transparency. This will prevent future issues with third party providers, printers, etc.