Literally translated, the long phrase 箸より重い物を持ったことがない means that you “never had to lift anything heavier than chopsticks”. Figuratively, this is a metaphor for never having experienced hardship or difficulties in one’s life. The image of the person who is referred as 箸より重い物を持ったことがない is that he or she has not experienced much physical work, including house work because their parents likely did everything for them.
箸より重い物を持ったことがない can actually be either positive or negative. The positive connotation is that the person received extra care, attention, and love from their parents. The negative connotation of course is that the person is inexperienced and does not understand the real way of living because everything was always done for them.
You may think 箸より重い物を持ったことがない is similar to 箱入り娘, but while 箸より重い物を持ったことがない can be used for both men and women, 箱入り娘 is only for women. However, using 箸より重い物を持ったことがない for men would be more insulting compared to when it is used to woman because it sounds like he has no physical strength at all!
箸より重い物を持ったことがない conjugates as an い-adjective as it ends with ない. For example, if you took extra care of your child and did everything for him/her, you can say （子どもに）箸より重い物を持たせたことがない with the causative verb of 持つ to create a meaning of “I have never let my child experience hardship in his/her life”.
Thanks to my parents, I have never experienced hardship in my life.
What? You can’t do such an easy thing? You don’t really have much life experience, do you?