The new layers of paint needs removing. You have 2 choices; scrape and sand or strip.
Scraping and sanding will work for most wood trim, including the door frames, but won't work for the doors. Using a paint stripper will also work for all surfaces but is a lot of work. Plus the doors could be very difficult due to the embossed wood grain.
To be honest I think it might be cheaper to replace the doors with new and maybe even the trim. From a painters point of view all of this work is expensive, installing a new pre-hung masonite door is fairly inexpensive.
If you get all of the new paint off and back to the original finish the prep is straight forward – prime everything. It is hard to believe that a little cleaning, maybe a little sanding, and the application of primer is all the painter needed to do but it's true. He probably put on 2 coats of latex paint anyways, if one of those coats was a good oil base primer there wouldn't be any problems.
I hope you didn't pay this person for low grade work.