Idiot Plot: The whole plot kicks off when Charles Darnay arrives in France, without telling his wife, his father, his boss about his whereabouts, all to protect his steward Gobelle. He arrives in France in the middle of a warzone, with virtually no documents despite being a former nobleman with none of the protections and connections that Mr. Lorry was able to foster at Tellson's branch in Paris. What makes it worse is that Darnay's actions very nearly becomes a Senseless Sacrifice with the person he wanted to rescue, Gobelle, being freed without his help at all and Darnay's actions endanger his family and leads to Carton's death.
Moral Event Horizon: No words are enough to describe the things that the Evremondes brothers did raping Madame Defarges's sister and murdering her brother and imprisoning Doctor Manette in the Bastille.
Also, the Little Seamstress whose only crime was being employed by a noble and who will never see her cousin again, but goes to her death with dignity and grace.
Additional Tear Jerker if you ascribe to it a bit of Alternative Character Interpretation. The Seamstress recognizes the switch and knows that Sydney is sacrificing himself. Some people like to think that they fall in love in the few moments they have together before their executions.
To some Madame Defarge's backstory is quite a huge tearjerker as well.
Values Dissonance: Cruncher's Domestic Abuse of his wife is more or less Played for Laughs, but to a modern reader it can come across as more disgusting than comedic. This has the side affect of making Cruncher less sympathetic than the author may have intended him to be.
The novel's rather negative portrayal of the French Revolution also grates on those who are more cognizant of its achievements note universal male suffrage, equal rights for Protestants and Jews, the no-fault divorce, a moderate amount of wealth redistribution and the abolition of slavery for the first time in the history of the world. Though this is typical since at the time, the Victorian establishment (which Dickens was critical of, but nonetheless supported) regarded these as "radical" not knowing that it would eventually become mainstream. That said, much of these achievements were abrogated in France itself in the intervening decades.
More specifically, the novel's famous description of the Carmagnole street demonstration portrays it as an example of crowd madness when in France it is seen as a beloved protest song against corrupt nobility.
Dickens's portrayal of the French Revolution was Fair for Its Day. At the time, many people saw the French Revolution as an unjustified burst of madness. In this view, the blame for the revolution lay with subversive secret societies, who had supposedly riled up the otherwise content French people. This is how conspiracy theories about The Illuminati got their start, by the way. While acknowledging that the French Revolution had had its problems, Dickens made it clear that the French people were being mistreated under the Ancien Régime and that the revolutionaries did have legitimate reasons to rebel.