The INSERT will just insert all rows and nothing special will happen, unless you have some kind of constraint disallowing duplicate / overlapping values (PRIMARY KEY, UNIQUE, CHECK or EXCLUDE constraint) - which you did not mention in your question. But that's what you are probably worried about.
Assuming a UNIQUE or PK constraint on (col1,col2), you are dealing with a textbook UPSERT situation. Many related questions and answers to find here.
Generally, if any constraint is violated, an exception is raised which (unless trapped in subtransaction like it's possible in a procedural server-side language like plpgsql) will roll back not only the statement, but the whole transaction.
Without concurrent writes
I.e.: No other transactions will try to write to the same table at the same time.
- Exclude rows that are already in the table with - WHERE NOT EXISTS ...or any other applicable technique:
 
- Select rows which are not present in other table 
- And don't forget to remove duplicates within the inserted set as well, which would not be excluded by the semi-anti-join - WHERE NOT EXISTS ...
 
One technique to deal with both at once would be EXCEPT:
INSERT INTO tbl (col1, col2)
VALUES
  (text 'v1', text 'v2')  -- explicit type cast may be needed in 1st row
, ('v3', 'v4')
, ('v3', 'v4')  -- beware of dupes in source
EXCEPT SELECT col1, col2 FROM tbl;
EXCEPT without the key word ALL folds duplicate rows in the source. If you know there are no dupes, or you don't want to fold duplicates silently, use EXCEPT ALL (or one of the other techniques). See:
Generally, if the target table is big, WHERE NOT EXISTS in combination with DISTINCT on the source will probably be faster:
INSERT INTO tbl (col1, col2)
SELECT *
FROM  (
   SELECT DISTINCT *
   FROM  (
       VALUES
         (text 'v1', text'v2')
       , ('v3', 'v4')
       , ('v3', 'v4')  -- dupes in source
      ) t(c1, c2)
   ) t
WHERE NOT EXISTS (
   SELECT FROM tbl
   WHERE  col1 = t.c1 AND col2 = t.c2
   );
If there can be many dupes, it pays to fold them in the source first. Else use one subquery less.
Related:
With concurrent writes
Use the Postgres UPSERT implementation INSERT ... ON CONFLICT ... in Postgres 9.5 or later:
INSERT INTO tbl (col1,col2)
SELECT DISTINCT *  -- still can't insert the same row more than once
FROM  (
   VALUES
     (text 'v1', text 'v2')
   , ('v3','v4')
   , ('v3','v4')  -- you still need to fold dupes in source!
  ) t(c1, c2)
ON CONFLICT DO NOTHING;  -- ignores rows with *any* conflict!
Further reading:
Documentation:
Craig's reference answer for UPSERT problems: