The terms libertarian and liberal do not have clear and precise meanings. They have been used differently in various historical contexts. Both words are connected to the notion of liberty and the political traditions trace their roots to John Stuart Mill's famous essay On Liberty. At the present point in US political history the libertarian tradition has an active following while the word liberal as it was used in the new sense is now linked to the much more frequently used term progressive.
Libertarians as they appear in contemporary politics have a high probability of being straight white males in comfortable economic circumstances. In short people who are the beneficiaries of privilege in most aspects of their lives. They typically advocate an absolute minimum of government activity. That gets focused both on the free and full exercise of their civil liberties, but also on not paying taxes to provide social support for people who are less well off than they are. There is usually a pretty definite divide between the libertarians and the people of the liberal/progressive tradition in terms of the economic issues. Most progressives are trying to hold onto what is left of the new deal. It is in the area of civil liberties that the relationship gets muddy.
The first amendment to the US constitution gives protections to free speech including political speech. The equal protection clause of the 14th amendment gives protection to the rights of minorities to be protected against illegal discrimination. This sets up a situation whereby two constitutionally protected rights are frequently in conflict. The strongly ideological libertarians take the position of absolute first amendment rights. Almost nothing should be allowed to restrict it. Maybe they would accept the classic example of yelling fire in a crowded theater when there is no fire as an allowable restriction, but not a lot more than that. The idea of legally defining a category of hate speech that infringes on the rights of women and minorities is utter anathema to them.
Most libertarians are not social conservatives. They don't like the notion of imposing personal moral codes on other people. They usually oppose restrictions on the private choices of consenting adults. Thus they are at times allied with some aspects of gay rights and women's reproductive rights in opposition to the social conservatives. However, when it comes to the issue of public expression of opinion that reflects sexism or homophobia a conflict arises.
We have a prime example of this conflict in the situation of Brendan Eich and his short lived career as CEO of Mozilla Corp. The general culture of the tech industry tends strongly toward libertarian values. It has a particularly high concentration of young white males. In general most of the major companies in that industry have adopted HR policies that are gay friendly. The women who have managed to get a foot in the door have typically found the atmosphere to be less welcoming. Issues around sexist language are not at all an unusual occurrence. So, it is a culture that is a mixed bag.
The immediate issue that aroused a protest to Eich's appointment as CEO was his $1000 contribution to the prop 8 campaign in 2008. As the story developed it also turned out that he has a history of financial support to far right political figures such as Pat Buchanan and Ron Paul. Added to the history was his refusal to state that his views on marriage equality might have evolved over the past five years and his suggestion that his position on the matter could be useful to the company in opening up the Indonesian market. Since he refuses to discuss his political views beyond that, we have to speculate as to how this fits into a general ideological position, but describing him as an ideological libertarian seems plausible. His position on marriage equality is decidedly in opposition to that of most of the libertarian culture of the tech industry. While there are numerous people claiming that he was fired because of his political activity, that is not entirely accurate. He resigned. He had continued in the position of CTO at Mozilla after the prop 8 campaign and there is no indication of any efforts to get him removed from that very well paying job. The problems only arouse when he was promoted to the position of CEO.
The right to political speech for Eich or anybody else does not exist in a vacuum of absolutism. We have here an excellent example of that right coming in conflict with the rights of other people. The people who chose to protest his appointment were exercising their legitimate rights to political speech. People make choices and choices have consequences. The right to take a public political stand does not include absolute protection form other people's reaction to that stand. The constitutional protection only extends to protection from governmental restriction.
The first amendment does not apply directly to private actors and organizations. There is a lot of discussion in this about protecting an open internet that allows for the expression of opinion. That issue is usually focused on government censorship. Most privately owned sites on the internet place various restrictions of speech. Daily Kos in no exception. DK offers an exceptionally high level of access with the ability to post articles without prior approval. However, like just about every other media site that provides the ability for readers to post comments, there are restrictions as to what is considered acceptable speech.
This is not a conflict that has a neat and tidy resolution. In nice polite middle circles there is a tradition of people avoiding topics that might become controversial. If you go to a dinner party or a business meeting it is usually prudent to avoid talking about politics, sex or religion. However, most people don't live all of their lives in such a well behaved bubble. For many people race, gender, sexual orientation and gender identity is a cause of frequent conflict and difficulty. Avoiding the subject is simply not an option.
Free speech is not an absolute right. It is subject to restrictions and has to be balanced against other equally important rights. I am not a libertarian.