[Disclaimer: I have not studied Buddhism as much as I would have liked. It may be that I am falsely representing certain Buddhist doctrines (and yes, I realize that they vary from sect to sect), but this is all to the best of my knowledge, and I believe it corresponds to the common grounds of the different sects.]
My first complaint about Buddhism is that it makes unsubstantiated claims about the nature of the universe: karma as a force exists and influences us (e.g., what form our next life will take is dictated by our karma); there is an endless cycle of birth and rebirth for an individual soul; there are six states of existence (Gods, demigods/Asura, humans, animals, ghosts, beings in hell), and then three higher realms of existence (the Realm of Desire, the Realm of Pure Form, the Realm of Formlessness); in some sense, the soul persists throughout its rebirths, even while the individual dies; and so on and so forth.
There is a wealth of these metaphysical assertions. Yet for all that, the Buddha never provides us with evidence or any good reason to support the above claims! What do we have instead? Mystic revelation. The Buddha, as he sat and meditated under the bodhi tree, apparently tapped into a deeper source of knowledge - a source inaccessible to the rest of us - at which point he discovered The Truth. Presumably, then, we must trust the Buddha's word that things really are as he claimed. We cannot experience The Truth As Revealed To The Buddha ourselves until we achieve a similar state of enlightenment (if we ever manage to), and there is no way to either vindicate or falsify his "Truth" otherwise.
If you are like me, that alone will not feel particularly satisfying. When they are of such an extraordinary nature, I find it difficult to take anyone's claims on faith alone. I imagine most Buddhists would adopt an "If you don't believe it, try it yourself" attitude, saying that I too would eventually experience The Truth if I got to that point myself. But, attaining nirvana is supposed to be incredibly difficult; and, there is no way to know that one is doing the right thing until one actually reaches the end (nirvana). We are told that one can spend one's entire lifetime (and more! So many more lifetimes) seeking enlightenment unsuccessfully. If we could conceivably waste the rest of our lives pursuing nirvana without success, should we not want a little more assurance than "You will see it once you reach enlightenment ourself"? Since we cannot prove that the soul exists or that reincarnation works as advertised, then for all we know we only have one life to live. Would it not be more worthwhile to spend this one life in other pursuits, striving for things we are reasonably likely to achieve in one lifetime?
Many or all religions suffer from this problem of unverifiability. So this is not anything particularly special by itself. At any rate, the Buddha himself discouraged speculation and arguments about metaphysical issues beyond what he laid out himself. So let me move on to reveal the more grievous problem with Buddhistic doctrines.
The four noble truths of Buddhism are as follows:
1. Existence is suffering.
2. Suffering is caused by desire/attachment.
3. It is possible to escape desire (and hence suffering).
4. The way to escape desire is through the Eightfold Path.
The the first truth - to exist is to suffer constantly - is pretty uncontroversial (isn't it?), but the second - suffering is caused by desire and/or attachment - might be less obvious. Is it really the case that all suffering results from desire? Well, when we look at examples of suffering, they are invariably accompanied by a desire to relieve that suffering in some way or another. Moreover, the ostensible "cause" of suffering (as understood by the sufferer) always stems from desire or attachment itself. For example, I may feel upset that a friend snubs me - this is because I wanted her to treat me otherwise than she did. I may feel hurt that my son dies, and this hurt derived from my feelings of love and care - my attachment - that I experienced toward him.
So, to cut to the chase, Buddha's strategy is pretty simple. We just found out what causes suffering: desire and attachment. So, let us rid ourselves of those things. The fourth noble truth points us toward the Eightfold Path, and we are told that practicing the Eightfold Path will lead to a gradual lessening of our fixation on transient things. The eventual aim of Buddhism is to realize that we are all nothing (or "no thing" - nothing we can comprehend); Buddha advocated the belief of anatman - that there is no self. Furthermore, everything changes and fluctuates, and so it is folly to become attached to anything and treat it as permanent. Permanence, for the Buddha, is an illusion.
What is so wrong with all of the above?
If there is no self (since my "I" changes from moment to moment), how can I really do anything, much less seek enlightenment? If there is no self, what precisely is being deluded and subject to suffering?
If all desires lead to suffering, what is the worth of compassion? Buddhism emphasizes compassion frequently, and Buddha himself remained in our plane in order to spread his words and alleviate our suffering. But how could he have been motivated to do so? At the point of enlightenment, he would have seen that all desires are pointless or harmful, including the desire to help others. In essence, I deny that Bodhisattvas (those who attain freedom from desire/attachment yet remain in the world anyway to help others) are possible.
Finally, let us suppose for an instant that the Buddha is wrong, that reincarnation is false, and in fact, there is no afterlife whatsoever. If so, should not death be the easiest way to achieve this sort of state - where one has no more desires or attachments, where one is free from suffering? Why do Buddhists not kill themselves, but for the threat of incurring bad karma?
And, on that note, is it even possible for a sentient, conscious being to still exist without desire? I would think such a being must want to remain free from desires and attachments in order to continually effect that state. The want to remain aloof from desires is still a want.
That is the true problem here: Buddhism entices us to accept its tenants in order to be free of suffering. Yet, that desire to be free from suffering was itself an attachment, an attachment to our own egos. At the least, some desires seem to lead us away from suffering - such as the desire to know The Truth. If so, it is not actually true that all desires or attachments cause suffering. Hence, Buddhist beliefs undermine themselves.