Its not that the Irken's dont have emotions - we've seen them be happy, angry, sad - it's just that they don't form emotional connections to other Irkens / living things.
I imagine Irkens could probably spend years in isolation, without contact with other lifeforms, without going insane, or suffering psychological damage the way a human would, so long as they had something interesting or productive to do (which is important for a species that sends invaders on long missions to other worlds on their own. Zim took six months to get to Earth, and just seemed annoyed by GIR's singing, and most Invaders wouldn't even have a SIR Unit with GIR's level of sapience.)
Even Zim is only ever seen forming real connections to his subserviant robot henchmen, like GIR and Minimoose, never with other Irkens or humans (GIR may not always have the intelligence/attention span to follow/remember complex commands, but he does consider Zim his master.) Even then, Zim's defectiveness stems more from his delusions of granduer, inability to follow orders, and short-sightedness to the point of causing unintentional destruction on a regular basis, rather than anything to do with emotional connections