How do I remove duplicate records in a join table in PostgreSQL?
In my experience (and as shown in many tests) NOT IN as demonstrated by @gsiems is rather slow and scales terribly. The inverse IN is typically faster (where you can reformulate that way, like in this case), but this query with EXISTS (doing exactly what you asked) should be much faster yet - with big tables by orders of magnitude:
DELETE FROM questions_tags q
WHERE  EXISTS (
   SELECT FROM questions_tags q1
   WHERE  q1.ctid &lt; q.ctid
   AND    q1.question_id = q.question_id
   AND    q1.tag_id = q.tag_id
   );

Deletes every row where another row with the same (tag_id, question_id) and a smaller ctid exists. (Effectively keeps the first instance according to the physical order of tuples.) Using ctid in the absence of a better alternative, your table does not seem to have a PK or any other unique (set of) column(s).
ctid is the internal tuple identifier present in every row and necessarily unique. Further reading:

How do I decompose ctid into page and row numbers?
How list all tables with data changes in the last 24 hours?
How do I (or can I) SELECT DISTINCT on multiple columns?

Test
I ran a test case with this table matched to your question and 100k rows:
CREATE TABLE questions_tags(
  question_id integer NOT NULL
, tag_id      integer NOT NULL
);

INSERT INTO questions_tags (question_id, tag_id)
SELECT (random()* 100)::int, (random()* 100)::int
FROM   generate_series(1, 100000);

ANALYZE questions_tags;

Indexes do not help in this case.
Results
NOT IN
The SQLfiddle times out.
Tried the same locally but I canceled it, too, after several minutes.
EXISTS
Finishes in half a second in this SQLfiddle.
Alternatives
If you are going to delete most of the rows, it will be faster to select the survivors into another table, drop the original and rename the survivor's table. Careful, this has implications if you have view or foreign keys (or other dependencies) defined on the original.
If you have dependencies and want to keep them, you could:

Drop all foreign keys and indexes - for performance.
SELECT survivors to a temporary table.
TRUNCATE the original.
Re-INSERT survivors.
Re-CREATE indexes and foreign keys.
Views can just stay, they have no impact on performance. More here or here.