A. You don't like the lecturer, do you? (Correct! This is now - in the present)
B. You DON'T like the lecturer, DID YOU? (Incorrect, "Don't" is present-today "Did" is past-yesterday)
But you might say: You DIDN'T like the lecturer, DID you? (now both parts are in the past)
C. You DON'T like the lecturer AREN'T you? (incorrect as the parts don't match)
But you might say: You ARE going to the lecture, AREN'T you? (see, the parts match)
D. You DON'T like the lecturer, DON'T you? (incorrect: only use the negative once, as in A).
Hope this helps,
Take care,
Josephine