In On the Face of It by Susan Hill, Lamb says that the gate is always open and he does not like closing things [Dialogues of scene 1]. But then, why does Derry open the gate panting (Scene 3)? What is the significance of this?
There in no need to suppose that there is a particular significance. Open does not only mean 'not closed' but also 'not locked' Oxford English Dictionary
Of a door, gate, window, etc.: having the movable part put aside so that there is access or passage through a wall, fence, etc.; not closed or blocked up; (also) unlocked. Of a doorway, gateway, passage, etc.: allowing access or view, free from obstruction; (also) not locked.
This leaves the options that the gate was 'shut' but unlocked, or that it was only open slightly such that Derry had to open it wider to enter.
Derry was panting because he had run back to Mr Lamb's house due to the argument with his mother and his desire to assure Mr Lamb that he was good to his word and was returning.