Convert Adobe Illustrator to Shapefiles:
Assumes you have a copy of Adobe Illustrator (or trial version) you must export as DWG file.
1) If necessary, sort the linework to layers based on the attributes you want the GIS data to have. The layer names will end up in a field "Layer" in the converted data, so be as descriptive as possible.
2) Curves in Illustrator must be converted to lines before export. Use Object>Path> "Add Anchor Points" command to add nodes to the curves (may need to invoke several times), and Object>Path> "Simplify" command to convert to lines.
3) Export to older version of DWG or DXF (R14 works).
4) View in ArcGIS, and export the data to a shapefile then use the Spatial Adjustment tool (does not work with a personal/file geodatabases)
5) Use the Spatial Adjustment toolbar in an edit session to match the linework to points on a known projection (the ArcGIS Help has quite a bit about this). You can match as many points as you wish, and you can choose the method ArcMap uses to align the data.
6) Export your data to a geodatabase feature class and set the projection.
Best Answer
As @Mapperz suggested, you can save a layer as a DXF from QGIS. The DXF can be imported into Illustrator easily enough, though you may have issues with scaling.
However, as Illustrator drawings are often set up as standard pages, it's probably better to create a print composer and save the page as either an SVG or a PDF.
If you're interested in purchasing a 3rd party program, I've used MaPublisher by Avenza before, and it's pretty good. It can import standard GIS files directly into Illustrator.