Sometimes the adhesive left over from vinyl tiling just stains the concrete. It may not be embedded in the concrete pores. However, before considering ANY paint coating, a simple porosity test should be conducted.
To run the test:
Let a few drops of water fall over an area of the surface on which adhesive glue residue is still visible. If the water soaks quickly into the concrete, the surface is porous and can be painted. If the water beads or remains on the surface, then an adhesive layer exists on the surface of the concrete. This will interfere with proper adhesion.
If the surface is not porous, any paint will have a hard time adhering. A concrete grinder will be required to remove all adhesive residue and to properly profile and prepare the concrete surface for painting.