How do I filter a pandas DataFrame based on value counts?

Use groupby filter:

In [11]: df = pd.DataFrame([[1, 2], [1, 4], [5, 6]], columns=['A', 'B'])

In [12]: df
Out[12]:
   A  B
0  1  2
1  1  4
2  5  6

In [13]: df.groupby("A").filter(lambda x: len(x) > 1)
Out[13]:
   A  B
0  1  2
1  1  4

I recommend reading the split-combine-section of the docs.