This is The Paradox of Tolerance, as mentioned in the comments, and it is a debated grey area.
Let's use your three examples, and let's observe that these all happened in the US. This constrains their utility as examples of liberalism, but the concrete examples are useful.
Free Speech and Tolerance means you will not be punished for your ideas within limits, and that your presence is tolerated. It does not mean we have to like you. It does not mean we need to publish your book or print your opinion piece. It does not mean you are given a megaphone. It does not mean anyone has to listen to you. It does not mean your words do not have consequences. It does not mean you get to yell whatever you like whenever you like at whomever you like.
Those limits generally are around when two person's rights collide and they must be weighed against each other. For example, the "clear and present danger" and later "imminent lawless action" US legal doctrines. Talking about what it would be like to yell "fire" in a crowded theater is protected speech and tolerated. Actually falsely yelling "fire" in a crowded theater is not protected, it puts people's lives in danger. Extending that a bit, a mob boss ordering someone to be killed is not protected.
The concept of Hate Speech is founded on a broadening of this idea when speech is intended to incite violence against a group. Much arguing has been around trying to define this grey area.
Free Speech is never a justification for your words. When your only justification for what you said is "free speech" you are admitting what you said is indefensible. Free Speech is a defense against censorship, not a defense against being a dumb-ass.
Chick-fil-A's president being fired for speaking against gay marriage
I can't find this particular instance, but there's plenty of cases of executives being fired for discriminatory public statements. This is a corporate board firing one of their executives because they feel they're bad for the business, and that includes their employees who need to have faith that their leadership is looking out for them.
Why would they be bad for business and shake confidence in their leadership? Chick-fil-A is a good example to dig into. As The Good Place put it, "There's this chicken sandwich that if you eat it, it means you hate gay people — and it's delicious!". Chick-fil-A is a multi-billion dollar fast food company who choose to donate to anti-LGBTQ causes and whose executives choose to speak very publicly against LGBTQ people. Emphasis on choose: nobody says a chicken company has to make political donations, nor that their executives have to go on talk shows and espouse their views.
In the US, corporations have an outsized role in their employees' lives. If you're an LGBTQ Chick-fil-A employee, watching the company president and the company itself publicly and actively campaign against your existence would raise questions about whether you will be treated fairly... at a chicken restaurant. These people have the power over your livelihood and your health care. It's an absurd situation, but here we are. That is not a good workplace environment.
The US has some odd quirks stemming from the 2010 Citizens United ruling which can be summed up as "money is speech" and "corporations are people". Therefore political donations are speech, and corporations are have the right to free speech, so corporate donations to political campaigns could not be limited. This allowed corporations to get deeply involved in the US political process.
The US also has some odd quirks around its ideas of Freedom of Religion. Burwell v Hobby Lobby in 2014 held that "closely held for-profit corporations" could just ignore regulations their owners had religious objections to. For example, the requirement to provide adequate health insurance to their employees, or the requirements against discrimination. They don't mean just mom and pop shops who don't want to bake cakes for gay people, the ruling was about a multi-billion dollar corporation whose decisions impact 43,000 employees and everyone who shops at their 1,000 stores.
This leaves US citizens with an ethical dilemma. Doing business with a corporation tacitly funds their political donations. Burwell v Hobby Lobby means this can be even more direct as the owners put their religious stamp on their employees. If you don't agree with their political funding, is it moral to purchase their products? Some deride this as "Cancel Culture", but it's an unfortunate consequence of the ethical dilemma Citizens United exacerbated. If money is speech, people will "vote with their dollars". This is the "chicken sandwich that if you eat it, it means you hate gay people". It's absurd, but a couple of questionable court rulings and here we are.
The ethical necessity of "voting with your dollars" means if companies decide to donate to political causes which their customers and employees find abhorrent, their customers can refuse to pay into this. This is not censorship - the company can continue to donate as it likes - this is the consequence of the company's choices. Chick-fil-A put their customers in this position, nobody forced them to. Tolerance, if it applies to a corporation at all, requires we allow Chick-fil-A to donate to whomever they want. Tolerance does not require that their customers turn a blind eye to this and indirectly donate to causes they disagree with.
Similarly, if the company executives decide to make their political positions known far and wide (nobody's forcing them to go on talk shows), people don't have to give them money, and their employees are right to be concerned about how the political positions of their leadership will affect them.
The Free Market says if Chick-fil-A's customers aren't happy, the company will fail. Their board is there to ensure the company succeeds. If their president endangers that, the board is entirely within their rights (and arguably responsibilities) to fire that person.
A NYT editor resigning after publishing an op-ed promoting the use of force to quell largely peaceful protest
The New York Times announced on Sunday that its editorial page editor had resigned after backlash from the public and the company’s own employees over a Republican senator’s op-ed that called for using military force against recent rioters.
As with Chick-fil-A, the New York Times is a business.
James Bennet was their Editorial Page editor. He was the guy in charge of handing out megaphones. He chose to give a megaphone to Senator Tom Cotton and approved his piece "Send In The Troops" in which a US Senator opined the military should be used against peaceful black protesters.
Publishing this upset many of their readers, journalists, and employees, particularly black employees. They questioned James Bennet's wisdom in choosing to broadcast this piece. It came out that Bennet did not read the op-ed, a basic part of his job.
This is not a free speech case. Tom Cotton was not censored. Nobody, not even a newspaper, has to give anyone a megaphone. In particular a US Senator has plenty of ways of making their opinion known. As for tolerance, a newspaper's customers can express their disappointment in who their newspaper is giving megaphones to by not paying for that newspaper. Intolerance would be to call for the Times to be shut down over who they give megaphones to.
The Times trusted James Bennet to decide who got to use the Times's megaphones, they trusted him with their integrity, and he was negligent. He angered their customers, caused issues with his fellow employees, and endangered the Times's journalistic integrity. As such, he failed to perform the duties required by his employment, and that is the reason he was asked to resign.
Removing statues of people that espoused inequality of human rights
Statues are not people, they are not granted free speech and tolerance. Statues are not speech. Statues are not history, but statues have a history. Statues espouse a historic viewpoint that these people are heroes. Statues are propaganda.
Public statues are government megaphones. They commemorate and celebrate people and events which we, as a society, decide reflect our values. Our values evolve over time, and so does how we choose to use that megaphone.
Confederate statues in the US perpetuate the idea that these are military heroes of the United States to be celebrated fighting a "lost cause". These statues ignore that they were anti-democratic insurrectionists who fought a war because they didn't like who got elected president and so they could keep people enslaved. They white-wash US history.
You don't learn history from a statue, you are reminded of it. A heroic Confederate statue reminds us that the US is OK with intolerance, racists, and people who believe other people are property. We are moving away from that. We are changing what we want to say with our megaphones.
If one wants to preserve these statues and their history, put them in a museum in their proper context. Explain how we came to make statues of people who tried to tear the country apart. Use them to talk about the Jim Crow era, the Civil Rights movement, and how they were used as symbols of oppression. How they were used to white-wash US history. That would be free speech and tolerance.