John Adams dismissed Timothy Pickering as Secretary of State in 1800. Adams first asked for his resignation, which Pickering declined to give, and then Adams fired him. There were tensions between the two men for a while; Adams found that Pickering and his colleagues in the cabinet were in being advised on policy by Alexander Hamilton, whom Adams despised. I've also read that Pickering publicly criticized Adams' diplomacy with the French during the Quasi-War, but I unfortunately can't find any examples of those criticisms right now.
On a side note, Adams also requested the resignation of his Secretary of War, James McHenry. Unlike Pickering, McHenry obliged Adams and submitted his resignation.
So I can only think of two examples: Andrew Johnson and Woodrow Wilson (sort of).
After the end of the Civil War, Lincoln's planned Reconstruction was going to radically change the South, and he accordingly selected a cabinet to do that. His assassination, however, made Andrew Johnson President. Johnson was from Tennessee, and Congress was wary that he would be lenient towards the South during Reconstruction. To prevent this, Congress passed the Tenure of Office Act, which would have forced the President to get Senate approval to remove a Cabinet-level officer.
Johnson was not a big fan of Lincoln's Secretary of War Edward Stanton, to the point of wanting to replace him with Ulysses S. Grant. Since Stanton would not retire his position, Johnson removed him while Senate was out of session. When the Senate reconvened, per the Tenure of Office Act, they voted on the removal, which they rejected. Johnson then tried to bring in a new Secretary of War before Congress began impeachment proceedings. In the end, Stanton resigned.
Woodrow Wilson fired a Post-Master General (not quite a Cabinet member, but considered to be of the same class as one). It brought on a huge lawsuit that, at least some what based on the constitutionality of the then-repealed Tenure of Office Act, said that the President could not outright fire cabinet members. The Supreme Court disagreed in Meyers v. the United States, saying that the President could fire any cabinet officer without the approval of any other branch of government.
Tl;dr: Johnson did, got impeached; Wilson fired an equivalent, got sued
I'll have to source this later because I don't have my books here...