There is a major problem in libertarian philosophy that has been ignored for far too long. Although the ideology has withstood attacks from many directions, I believe that this argument has the potential to destroy it. The Achilles’s heel of a libertarian is the fact that they are logically required to approve of suicide. 
I am not just talking about euthanasia. Any rational person who wishes to end their life, whether it is an 80-year-old with heart failure, or an 18-year-old after a bad break up, must be allowed to do so. This conclusion is the direct result of the libertarian’s doctrine of self-ownership, which says that a person owns their body. In contrast, most philosophies say that people have two independent sets of rights: property rights and body rights.
It is hard to stress just how fundamental self-ownership is to libertarians. The first section of Stanford Encyclopedia of Philosophy’s article on libertarianism is about self-ownership. How self-ownership leads to this rather extreme conclusion is fairly simple. A person may do with their property as they wish, assuming that everyone directly affected by the action consents. If one wishes to drop their new cell phone out of their window, they have every right to do.
If you raise the argument that doing so is foolish, wasteful, or in any other sense, immoral, the libertarian will simply counter that those are reasons for why someone should not exercise their right to destroy their own property, not an argument that such a right does not exist. If someone has a right to do an action, they can do so even if it is a bad idea. 
Replace ‘cell phone’ with ‘body’ and the proof is complete. While most philosophers would object to such a substitution because different rights govern one’s property and body, the libertarian cannot. If the body is just another type of property, then whatever rights I have regarding my material objects, I have over my body.
This is actually the same line of reasoning that is regularly used by libertarians. If I have the right to mail letters anywhere I want, I must have the right to travel wherever I want. If I have the right to put whatever types of fuel I want into my car, I must have the right to put whatever types of drugs I want into my body. If I have the right to sell my goods to whomever I want, I must have the right to sell my body via prostitution. Indeed, many libertarian institutions, such as the Cato Institute and the Libertarian Party of America, support open borders, ending the war on drugs, and deregulated sex work. 
You may have noticed a loophole though. Libertarians say that it is immoral for someone to interfere with the actions of any rational person. This means that it is acceptable to restrict the liberty of people unable to rationally evaluate what is in their best interest, such as the insane. A libertarian could claim that suicide is inherently irrational, and therefore anyone wishing to commit it must be mentally ill. Claiming that the suicidal are mentally ill is not particularly controversial, so this seems like a good solution, but it does not hold up under scrutiny.
The problem with this proposal is that mental illness is not enough justification to limit someone’s freedom. The person must be insane, or incapable of rational thought. If you modify the position to say that anyone who is suicidal must be incapable of irrational thought, you reach three new problems.
The first is empirical; any trained psychologists will tell you that many suicidal people are not insane. There are also plenty of historical counterexamples, the most famous of which would probably be Socrates (although he was sentenced to death, Socrates refused to escape from Athens despite having many opportunities).
Secondly, this proposal outright forbids euthanasia, regardless of the circumstances. I will not go into details, but I am sure you can come up with situations on your own where a rational person would want to stop living.
The third problem is also the biggest; people who are insane cannot be held accountable for their actions, morally speaking. This follows from Immanuel Kant’s famous conclusion that “ought implies can.” In other words, a person is only responsible for what they can control. Someone not in control of their mental faculties cannot be held responsible for what they do.
Here is where things get really uncomfortable. If all persons who commit the act of suicide are insane, and no insane people can be morally responsible for their actions, then suicidal actions cannot be immoral. This includes suicidal actions that harm people other than the one committing suicide, such as suicidal terrorist attacks. Someone claiming that suicidal people are insane are logically required to admit that most terrorist attacks are not immoral. Like I said, this is really uncomfortable. Strictly speaking, this position is valid from the point of view of formal logic, but I highly doubt anyone will be willing to take it.
There is one other way the libertarian might try to slip out of this. The libertarians say that if an action harms people who do not consent, then the action is immoral. Because suicide causes severe mental stress to the friends and family of the victim, the act is immoral. The problem with this logic becomes apparent when you substitute other actions for suicide. What if, instead of committing suicide, someone wanted to convert to a different religion, have casual sex, or start using drugs? According to this line of reasoning, if these actions would cause mental stress to other people, then the government can step in and interfere. Since libertarians would unequivocally say that government interference in such a way is wrong, they are unable to also say that mental stress is a harm significant enough to limit someone’s freedoms.
There is really no way for libertarians to get around this problem. Because they accept the concept of self-ownership, they are forced to accept that people have a right to suicide. From this, it follows that actions taken to stop someone from committing suicide are immoral; suicide help lines are coercive and doctors treating suicide attempts should go to jail for assault.
I believe that this issue would be a deal breaker for many libertarians, which is why it is hardly discussed. The only libertarian, to my knowledge, who has addressed this problem is Dr. Szasz, and he actually agreed with the conclusions I laid out. Famously, he took his patients of off medications prescribed to combat suicidal tendencies even after this resulted in some of them preforming the act.
This problem strikes at the very heart of libertarianism. No matter how much you like small government or laissez-faire capitalism, you cannot use libertarian arguments to justify them without also committing yourself to a truly unacceptable position.