One way to gain an understanding of past and future lives is to examine the process of sleeping, dreaming, and waking, because this closely resembles the process of death, intermediate state, and rebirth. When we fall asleep, our gross inner winds gather and dissolve inwards, and our mind becomes progressively more and more subtle until it transforms into the very subtle mind of the clear light of sleep. While the clear light of sleep is manifest, we experience deep sleep, and to others we resemble a dead person. When the clear light of sleep ends, our mind becomes gradually more and more gross and we pass through the various levels of the dream state. Finally, our normal powers of memory and mental control are restored and we wake up. When this happens, our dream world disappears and we perceive the world of the waking state.
What happens when we die?
A very similar process occurs when we die. As we die, our winds dissolve inwards and our mind becomes progressively more and more subtle until the very subtle mind of the clear light of death becomes manifest. The experience of the clear light of death is very similar to the experience of deep sleep. After the clear light of death has ceased, we experience the stages of the intermediate state, or bardo in Tibetan, which is a dream-like state that occurs between death and rebirth. After a few days or weeks, the intermediate state ends and we take rebirth. Just as when we wake from sleep, the dream world disappears and we perceive the world of the waking state, so when we take rebirth the appearances of the intermediate state cease and we perceive the world of our next life.
The only significant difference between the process of sleeping, dreaming, and waking and the process of death, intermediate state, and rebirth is that after the clear light of sleep has ceased, the relationship between our mind and our present body remains intact, whereas after the clear light of death this relationship is broken. By contemplating this, we will gain conviction in the existence of past and future lives.
String theory shows that reincarnation is fully possible. If we truly have an infinite amount of parallels, the conscious mind (at least of some) must be going to them. It is not such a far fetched idea. It is actually one that confirms many world religions, as well as disputing others. But who knows? Heaven and hell could be parallels.
With the unknown, anything is possible. I recently watched I Origins, a movie which delved into this possibility. I do believe that there is a lot yet to be known, and it is better to be open-minded than closeted. We are not really sure if heaven is real either, all we have are books and religion to instate the realness of heaven, and yet it is not enough. We do not truly know, we are limited.
I don't see how reincarnation can work if the total population of living things is not at a state of equilibrium throughout all of eternity. If every being is just reborn in a new one, we shouldn't have any "new" souls born. We would all just be recycled bodies. There are possibly ways where you could move around it, but these ideas do not fit within the general framework of hinduism nor buddhism, where the universe is in perpetual balance.
Choosing no, just because we don't have any evidence of it existing.
Although, with that logic, one could argue "then, heaven cannot exist, because we don't have evidence of it existing". Yes, there have been 'visions' or 'dreams' of people being in heaven, but they could have just been actually dreaming, and them possibly being delusional, and thus, not providing sufficient evidence on the existence of heaven.
On the other side of an argument, I'll have to agree with GKY's statement "with the unknown, anything is possible". Then, one could rebut that "since heaven is unknown, it could exist'. Going a little deeper, no firm conclusion can be drawn, sadly. There's the same evidence for the existence of heaven (or and afterlife) and reincarnation. Which you go with can be a product of many factors, such as your religion/beliefs, personal ideals, or which you find to be more reasonable.
In the end, I just picked "no" because I'm a Christian, and I didn't want anyone to feel pressured into choosing Yes, because that definitely happens to some people.