We live in a high-rise apartment, and the shower room cinder-block wall is covered by sheet rock. The wall separates our apartment from our neighbors' apartment. Anyway, I noticed that the blocks above the drop-ceiling seemed to have no vertical joints. I imagine the hidden part of the wall below the drop-ceiling is the same. I can almost put my small finger into the spaces. Is this type of block-laying common? Or should we think of looking for a new home?
No this is not common. In some areas it might even violate code as there might not be fire separation.
However to say does this mean anything structurally... who knows without full details of the building. Unless you live in a third world country I doubt that there would be a building with more than 10 stories that relies on cinder blocks at all for support. Most interior walls in high rises are made to be taken down and moved if needed. Most buildings rely on exterior walls and key points usually in the center square of the building.
Why is there a gap in yours? Maybe it is for expansion - but then this should have fire rated caulking. Maybe the workers were lazy and knew it would be covered... who knows. Should you be worried? No.
Note: You should not be worried about the building falling over. However if there is smoke coming in yes you should be worried about both your health (air quality) and fire will spread faster (well I would gather that it isn't the fire itself, it seems the smoke from a fire could spread and be hazardous easily in your situation).
It's tough to tell without pictures or a more specific description, but this isn't necessarily a problem. There are a few reasons there may not be mortar in every joint, like crack control, thermal expansion, or seismic separation. If things otherwise look like they're in good condition I would say it's probably deliberate. However if you see crumbling mortar or cracks in the masonry units (a.k.a. "cinder blocks") themselves, there may be deterioration or poor construction.
If it's bothering you I would bring it up with your landlord or condo association. But personally I would frame the conversation more as "I just noticed this and wanted to make sure it's OK", not "the building is falling down!".