What is the meaning of 放 fàng in this idiom?
'To put; to place' and 'to release' are not entirely antonymous. The original logic in the character 放 is that you release something from your hand such that it is placed on a receptacle.

It can be literal, as in 他放下錢便走了 ('He left after putting down some money'); or figurative, as in 別放在心上.

To be precise, you are advised 'not to put something on your heart'. Let us hypothesise and say 她的死 (her death) is the object, as in 'don't take her death to heart'. We can say 別放她的死在心上, but it sounds better if we use the 把-construction, as in 別把她的死放在心上. The object is thus usually fronted in the sentence, which may be a source of your confusion.