How to remove an element from a col with a CSV of ints in PostgreSQL 9.2?
Proof of concept for regular expressions:
SELECT trim(regexp_replace(',' || '3,4,5,13,14' || ',', ',(4|5)(?=,)', '', 'g'), ',');

Returns:
3,13,14

Pad commas to cover corner cases at begin and end. The regular expression ',(4|5)(?=,)' explained:
, ... literal comma
(4|5) ... two branches: either 4 or 5
(?=,) ... positive lookahead: next character is a literal comma
Would need a functional trigram index on (',' || allowed_types || ',') for good performance with big tables (only slightly bigger than a regular index due to the added commas). Details:

How is LIKE implemented?

Or, with a more sophisticated regular expression, you can work with the original column and a trigram index on just (allowed_types):
SELECT ltrim(regexp_replace('3,4,5,13,14', '((,|^)(4|5))(?=,|$)', '', 'g'), ',');

dbfiddle here
But I expect the first solution to be faster: complex regular expressions are more expensive.
The according UPDATE
UPDATE tbl
SET    allowed_types = trim(regexp_replace(',' || allowed_types || ',', ',(4|5)(?=,)', '', 'g'), ',')
WHERE  ',' || allowed_types || ',' ~ ',(4|5),';  -- here, plain comma is good

What I would really do:
Update to a current version of Postgres (pg 9.2 reaches EOL Sept 2017) and probably use a normalized 1:n design.