How to remove asphalt used to repair concrete
This would be easier to answer with some pictures of what you're dealing with.  But I doubt asphalt would have been used to fix chips and gouges.  It's more likely that the concrete broke all the way through, or was damaged during the driveway work, leaving holes or a full-depth ragged edge.  

A concrete repair won't bond well to asphalt.  Trying to surgically cut it out, or soften it with heat and scoop it out, will leave a mess on the concrete surface you want to bond to.  And with heat, you're likely to get stains from hot asphalt on other concrete spots as well.  It will also be a long, messy job.

Even if the asphalt patching is thin and you are planning a concrete surface patch, you can run into separation and wear problems.  Concrete products are cheap, and it will be faster, easier, and longer-lasting to just cut out the problem area and replace it.  I would use a concrete saw to make a clean cut and remove the asphalt-tainted material.  Then repair the hole. 

But if the current asphalt repair is solid and clean, I'd think twice about removing and patching it just because you think paint won't stick.  If you scuff up the surface and use a good primer, it may work well enough, and the difference in material may not be noticeable under the paint.  

If there is a lot of texture in the asphalt, grinding the surface and then filling any indentations with the right material, like an epoxy, could give it a smooth surface for paint.  Even if the paint wears off, slapping on another coat in a small area every few years would be a trivial job, and way less work than redoing a repair that isn't broken (if that's the case).