Why not paint the beams (all 3 sides) their own color. Glossy Black would really pop.

It sounds like you are trying to make the beams "disappear" into the ceiling. If they are structural and are unpainted, leave them that way. You will want to clean them really well at the very least. If they are already painted, paint them the color of the ceiling. If they are not structural, think about removing them so you can add crown moulding. :)

I always see them painted white like ceiling in these apartments. I can't believe I can't find photos on Houzz of beams painted same color as walls, or even their own color. They are structural in that coop will not let us alter them. We don't want to cover them b/c we like ceiling height. Seems so boring to paint them same as ceiling, but that appears to be what EVERYONE does. Can anyone point me to photo where beams are painted same color as walls or even their own color?

Here is just a few of many I found on house.

Type in painted ceiling beams and many photos will come up.