This is based on opinion. Please don't list it on a work's trope example list.
Tearjerker / Liberty's Kids
The death of Nathan Hale. Even James cries in that scene.
United We Stand: A mob tars and feathers a sailor named Mr. Parker, and James, thinking that the man got what he deserved for being unpatriotic, joins the crowd to laugh at the poor man's misfortune. When he expresses a desire to add an article on the event to the paper, Moses decides that James has to see what became of Mr. Parker. At the doctor's, James is treated to a rather graphic description of what happens when you're tarred and feathered followed by an equally explicit description of the side effects of the tar removal (namely the peeling away of skin and the subsequent infection of the resulting wounds), taken into the room to see the sailor covered in bandages and tearing up in pain, and then told that the tears cause an unending cycle of pain because the salt in the tears only makes that man hurt more when they leak into the wounds. James starts crying.
Moses: I brought the young man I mentioned earlier. Can he ask you a question or two?
What makes this even worse is it's unclear if he survived or not.
Two words: Cousin Tom. Sarah's tears during that scene say it all. Not to mention her narration afterwards:
Sarah: Oh, mother, Tom's death seems to me a sign of awful things to come. This conflict is not at all what James thought it would be. It won't be simply contest of ideas. It promises to be a contest of arms. It promises to be everything I had prayed we would avoid.
The real-life fate of Cornstalk and his son in the episode The New Frontier. They're detained at Ford Randolph, even though Cornstalk's tribe was neutral, and then they're wrongfully executed. Sarah is so horrified by what happens to them that she outright leaves the colonies and returns to England.
The reactions of the other main character's to Sarah's departure in the same episode can be seen as a Tear Jerker. James is so upset that he refuses to say goodbye to her, and is still somewhat bitter when she returns three episodes later. Henri, like some viewers, perhaps, is in silent tears during the scene. Walter Cronkite's solemn narration as Dr. Franklin really helps, as well.
The very early scene where the group has met Phyllis Wheatley and Henri reveals his backstory as a slave: his parents died on the trip to America and the captain forced Henri to pay off their debt. It's obviously a painful memory given the bitterness in Henri's voice and Sarah's disgust is quite apparent (albeit subdued) in her declaration of "How dare he...". Henri's response to that ("I belonged to him.") doesn't help matters and it positively hurts to see the flashback of Henri being terrified of James and Moses because he thought they wanted to hurt him.
James' backstory is that he was orphaned as an infant when his parents died in a fire caused by a lightning strike. He sought out Benjamin Franklin when he was old enough, but presumably lived on the streets or in an orphanage until then.
John Paul Jones mourning the loss of the Bonhomme Richard after capturing the Serapis. You can hear his voice break as his beloved vessel sinks beneath the waves.
John Paul Jones: She was a cast-off ship with a cast-off crew, Ms. Phillips. Yet she beat the best. And....she was mine.