Why is it said in Vedanta that you are not the doer of any action?
"Now if I am not the doer and reaper of any action what's the significance of law of karma?"
The law of Karma has significance only if one considers himself/herself as the doer. In reality, neither Karma exists, nor the result of Karma, nor the doership. That's the reason, whatever good or bad deeds one does, wouldn't go beyond the material world.
BG 5.14 - Neither "Doership" (Kartutva) nor "Actions" (Karma) nor "Reactions" (Karma-phala) of the people are created by the Omnipotent; But only their nature [3 modes] pervades.
BG 5.15 - The Omnipresent neither accepts anybody's sin nor even virtue. Knowledge remains covered by ignorance. Thus the creatures become deluded.
"Lord Krishna says what is happening with us is the result of our past actions."
Krishna says that, depending on good/bad actions, "I" will put that being in good/bad life. In other words, because a being wants good/bad life, he/she 'does' good/bad actions and reap good/bad fruits.
In brief, the supreme Brahman merely satisfies the wanting (desires) of all beings.
BG 13.22 - Purusha being seated(situated) in Prakruti, experiences the [sattva, rajas, tamas] modes born of Prakruti. Contact with the these modes, is the cause of its births in good and evil wombs.
"So, shouldn't we be taking responsibility of what we are doing instead of saying that we are not the doer?"
It's upto individuals. If one wants to enjoy in the worldly matters, then one would automatically take up doership. If one is disinterested in worldly matters then automatically, will surrender the doership.
"Am I the doer of my actions or not?"
Either I believe "I am the doer" OR I know "I am not the doer".
Because, everything is predestined.
Those who are interested in the worldly matters, would themselves accept the illusion of "doership".
BG 3.27 — All actions (karma-s) are enacted in Prakruti by  modes (guna-s). Bewildered with identity, the self(Atma) believes "'I' am the doer".
Those who have realised the predetermined nature of the world, would discard the doership themselves.
BG 5.8, 5.9 — "'I' certainly don't do anything" is thought by knower of elementary (tattva gyAna) - even while seeing, hearing, touching, smelling, eating, moving, dreaming, breathing, talking, discarding, accepting, opening, closing; -- "Senses only are acting in the subjects of senses" - is understood.
Think of this life as a cricket match. Hitting balls & getting runs OR throwing ball & getting OUT -- are Karma-s & Karma-phala-s respectively -- the rules. "Veda"-s are the guidelines for everyone on how to make this win/lose game a gentleman's game.
Gita simply says that, the whole match is spot fixed!
The spectators who want to enjoy this fixed match, would illusion themselves as of the match is not fixed. They will follow the actions of the player of their liking.
The disinterested people will have 2 groups:
- "Enlightened" -- They will sit in the match, but indifferent to all the happenings
- "Liberated" -- They will leave the stadium
Note: “The question: Why is it said in Vedanta that you are not the doer of any action?” is licensed by Stack Exchange Inc (https://hinduism.stackexchange.com/); user contributions licensed under CC BY-SA.