Does Advaita believe in reincarnation?
To answer simply, Advaita is not based on any single soul – there is no soul and not-soul at that ultimate understanding – there is only one non-dual sat chit and ananda. Existence-consciousness-ananda cannot be divided. If one sees divisions they are only apparent and not real. If one takes the apparent as real, then all others factors become as real.
Jiiva (soul) itself is a notion and when that notion is taken as real – all other problems become as real as jiiva. Hence reincarnation and transmigration of soul are all real in that frame of reference.
Look at this way: gold, iron and copper look different if these difference as taken as real. They can exist in different forms – now as ring, now as bangle, now as chain, now as bracelet – gold undergoing transmigration or reincarnation into different forms.
If one understands that all are nothing but electrons, protons, neutrons etc., then from that perspective gold, iron, copper are just bunches of electrons, protons and neutrons – which themselves are nothing but energy-states.
I can understand as a scientist they are all one – yet I can transact in the world taking gold as different from iron and copper. Transactions are done at one level while understanding is at the ultimate level – there is no confusion if one understands correctly. I know that the sun neither rises nor sets but I can still appreciate the beauty of sunrise and sunset.
That is advaita in spite of dvaita – that is no incarnation in spite of reincarnation. Karma (action) is at the transactional level. At the absolute level I realize that I am never a kartRRi (doer). That is Advaita. Advaita in spite of dvaita.
More Info: Advaita Vedanta of Shankaracharya