My wheels are slightly out of true, about 1.5 mm at worst part. It doesn't bother me when riding and I don't feel it, but will this hurt or damage the wheel if I keep going on, or should I keep going until it starts bothering me?
It won't present any immediate danger, however it became untrue for a reason. If the spoke tension has changed more than likely it will get worse over time and riding it while out of true will most certainly expedite that process. Other spokes will lose their tension and could possibly break resulting in a higher repair bill, or possible injury.
You will not immediately damage the wheel or rim but continuing to ride on it will only make it worse. It will also put stress on other aspects of the bike as well and your braking - if rim brake, will likely be slighty less effective.
All of these would be after a while though, so while there is no immediate danger it will get worse the more you ride. Hitting a curb with poorly tensioned spokes could cause one to "pop"