Albert Barnes Commentary Psalms 137:5

"If I forget thee, O Jerusalem, Let my right hand forget [her skill]." — Psalms 137:5 (ASV)

If I forget thee, O Jerusalem - The meaning here is that to sing in such circumstances would seem to imply that they had forgotten Jerusalem, that they were heedless of its sorrows and did not care that it was desolate. The remembrance of its calamities pressed hard upon them, and they could not do anything that would seem to imply that they had become heedless of the sufferings that had come upon their nation.

One will not rejoice when a wife or child lies dying, or on the day of the funeral, or over the grave of a mother. A joyous and brilliant party, accompanied with music, feasting, and dancing—when a friend has just been laid in the grave, when the calamities of war are widespread, when pestilence is raging in a city—we feel to be untimely, unseemly, and incongruous.

So these captives said it would be if they should rejoice while their temple was in ruins, while their city was desolate, and while their people were captives in a foreign land.

Let my right hand forget her cunning - Let my right hand forget its skill in music—all its skill. If I should now play on the harp—as indicative of joy—let the hand that would be used in sweeping over its strings become paralyzed and powerless. Let the punishment come where it would seem to be deserved—on the hand that could play at such a time. So Cranmer held the hand that had been used in signing a recantation of his faith in the fire, until it was burned off and dropped in the flames.