〜にしのびない
can't bear to (out of pity or attachment)
Verb dictionary form + にしのびない (に忍びない)
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にしのびない means you can't bring yourself to do something because it would feel cruel, sad or wasteful: まだ使えるので、捨てるにしのびない, "it still works, so I can't bear to throw it away".
The verb is usually an action that hurts something you care about: 捨てる, 処分する ("dispose of"), 断る ("refuse"), 見る, 聞く. It's about compassion or attachment.
忍ぶ means "to endure", so it's literally "can't endure to". Compare にたえない (N1), "too awful to (watch)", which is about how unpleasant something is, not about pity.
It's formal and a bit literary.
Examples
まだ使えるので、捨てるにしのびない。
It still works, so I can't bear to throw it away.
被災地の様子は、見るにしのびないものだった。
The scenes in the disaster area were heartbreaking to see.
長年使った机を処分するにしのびない。
I can't bring myself to get rid of the desk I've used for so many years.
彼の必死の頼みを断るにしのびない。
I can't bear to turn down his desperate request.
その事故の話は、聞くにしのびない。
The story of that accident is too painful to hear.