Libertarianism could be understood, in the most basic interpretation, as the view that property rights are of fundamental value. Thus understood, libertarians hold that individuals own themselves, and so may not be used or damaged without their consent. But we can distinguish between Right and Left-wing libertarians. Most libertarians are right-libertarians: they hold the unprincipled view that natural resources may be appropriated by some so as to leave none left over for others. This is not a viable position, for reasons explained in the linked post. Left-libertarians, by contrast, hold that all individuals have an equal right to natural resources. Ideally, they would begin (emerging from the 'state of nature') by giving everyone an equal share of natural resources, and then letting voluntary exchanges in the free market continue things from there.
But even this improved version of libertarianism fails, for reasons I have explained before. Any allocation of absolute property rights would allow the owner to destroy the resource in question, thereby depriving future generations of their fair share. Unless, that is, those shares were reserved in advance, but there would not be enough resources to split into indefinitely many positive shares. That's a problem regarding the origin of property rights. But even if resources could (somehow) be initially allocated in a fair manner, just transfer may be impeded by ignorance or accident, so there's no guarantee that market transfers will "preserve" the justice of the distribution. They might also cause some people to acquire an unacceptable amount of power over others. These objections point to internal problems: the libertarian principles of initial acquisition and just transfer offer an imperfect treatment of property rights.
There are also external objections which can be made, by pointing out that libertarianism is just fundamentally wrong-headed. Property rights are not of fundamental value. It's ludicruous that anyone should hold them to be more fundamentally important than human flourishing. We should instead take human welfare as our ultimate value - as utilitarians do - supporting whatever social/political institutions would be most conducive to this end. (This would no doubt lead to some form of conditional property rights, and probably redistributive taxation, rather than the property-rights absolutism of libertarianism.)
I have so far assumed that libertarians take rights as fundamental. They might instead claim that freedom (understood in the 'negative' sense of freedom from interference) is their fundamental value. But that also fails. Firstly, it fails on its own terms: enforcement of property rights is a form of coercive interference, and a reasonable resolution of the resulting "conflict of liberties" will lead to significant welfare rights for the poor. Secondly, it also suffers from the external objection that it is fundamentally misguided: what really matters is not merely freedom from interference, but rather, substantive freedom.
So, even left-libertarianism is, I think, quite misguided. Nevertheless it's interesting to consider what it involves in practice. Some left-libertarians argue for quite substantial taxation, to bring us closer to the ideal of 'equal initial resources' (e.g. for the next generation) that they believe is required by justice. Hillel Steiner offers three forms of "just tax" that can be rightfully imposed on property owners to achieve this end. The first and main one has already been mentioned: rectification for illicit appropriations of natural resources. The second is from bequests: dead people cannot own things, so nor can they transfer (bequeath) ownership rights to their descendents -- that's a legal fiction that ought to be got rid of. The property of the dead thus reverts to its natural status of common ownership, and can be distributed accordingly. (Realistically, I don't see that this would help much in practice: people would just make sure to transfer more of their property before they died.) Third, Steiner argues that genetic information is a natural resource that people (illictly) appropriate for procreative purposes, and so they must pay recompense for this too. That's rather silly though. Even if genetic info is in some sense a "natural resource", its use does not infringe upon the liberty of anyone else - it does not prevent them from using it too - so there are no grounds for recompense, unlike the first case.