Sheikh Muhammad bn Saalih Al-Uthaymeen (may Allaah have mercy upon him) said:

When you want to kill a rat, and killing it is Mustahhab (encouraged), then kill well. Kill it with what will take its life immediately, and do not hurt it. And from hurting it, is what some people do where they put down something adhesive to which it will be glued, then they leave it to die of hunger and thirst. And this is not permissible. When you put this adhesive catcher (in place), it's incumbent on you to repeatedly check and watch out such that when you see anything glued to it, you kill it. 

But for you to leave this adhesive catcher for 2 or 3 days and the rat is glued to it and dies of thirst or hunger, it is feared that you may enter hellfire for that, for the prophet, peace and blessings be upon him said: A woman entered hell because of a cat which she imprisoned until it died, she neither give it food nor did she free it to eat from the vermin of the earth. 

[Sharh Riyaad As-Saalih (4/596)]