Plants DO NOT feel pain.
To feel pain you need to have a central nervous system which all plants are lacking.
Also, from an evolutionary perspective a plant feeling pain would be useless as they lack the ability to escape from prey; a key purpose of feeling pain.
Plants do have physiological reactions (such as a carrot bruising) but that's due to the presence of oxygen and a withdrawal from their nutrient dense environment.
They do not feel pain.
That is a perfect answer. I was going to reply and give my own, considering for the past 6 weeks ive been studying plants in Biol in Uni (hate it hate hate it) but yes perfect answer!
Plants need a nervous system. So just to update everyone else, Prokayotes (Bacteria and Archea), Protists (Algae, yeast, plantkon, lots of random stuff) and Fungi and plants ALL DONT FEEL PAIN. Out of all the domains, and kingdoms only Animalia feel pain.Some animals dont feel pain, but they are very very simple organisms. Not your common sheep, lobster, fish, chicken or cow.
EDIT: I just double checked this too:
Pain motivates us to withdraw from potentially damaging situations, protect a damaged body part while it heals, and avoid those situations in the future. It is initiated by stimulation of nociceptors in the peripheral nervous system, or by damage to or malfunction of the peripheral or central nervous systems.
Its from Wikipedia, but still reliable. Plants do not have nociceptors, peripheral nervous system or central nervous system. BELIEVE ME! I have studied them literally INSIDE AND OUT. (ARGHHHH!!!)