tin man, not tin robot

tik-tok didn't turn to stone because he was a robot. the tin woodsman was not a robot, just a man made of tin.

Tinman was originally alive

In the books, the Tin Woodman was originally a (living) human person named Nick Chopper, who was fitted with a tin limb each time he accidentally amputated one (he was cursed). Everntually he became all tin, but one might conjecture that the tin man is infact alive, and while immune to many forms of abuse, not immune to a curse that affects the living.

The tin man is alive

If you would have listened to the part where Dorothy is in the dream building at the start she tells the man that the tin woodsMAN's axe was enchanted so he cut off all his body parts until he was all made of tin. You may say he would be dead after that, but this is Oz where things are magic so he is a man. Not a robot like tick-tock.

YES HE IS

If you actually paid attention, the tin man is/was alive. He was a real woodsman who eventually chopped off all of his body parts accidentally, including his head. I know that this is unrealistic, but you have to remember that it is a fantasy story.