ずにはいられない: a reaction one cannot suppress
Describe an action that emotion or a stimulus makes difficult to restrain. Check the negative connection and distinguish an internal impulse from external necessity.
An action that is hard to hold back
その話を聞くと、笑わずにはいられません。 means that hearing the story makes laughter impossible to suppress. It is not a prohibition against laughing, but a natural reaction.
困っている人を見ると、声をかけずにはいられません。 says that seeing someone in trouble makes the speaker unable to resist speaking to them. The action arises from a strong feeling or impulse.
Forms and the scope of negation
Remove ない from 笑わない and add ずにはいられない. する becomes せずにはいられない, so 感心せずにはいられません means being unable to help feeling impressed.
ないではいられない can occur in the same kind of context. 泣かないではいられません means being unable to keep from crying. Read both negatives together to identify the action that is hard to suppress.
An obligation differs from a reaction
規則で報告しなければなりません。 imposes a duty to report under a rule. 報告せずにはいられません。 describes an irresistible urge to report, so it is not a simple replacement in a statement of regulations.
In a report or recollection, 笑わずにはいられませんでした says laughter could not be suppressed at that time. 聞くと笑わずにはいられない can instead describe a recurring reaction, distinct from one past event.
Quick check: Does この写真を見ると、故郷を思い出さずにはいられません。 tell someone to forget their hometown?
Show the answer and why
No. It means that seeing this photograph inevitably brings the hometown to mind. The reaction 思い出す is difficult to suppress.