Regular expression in PostgreSQL LIKE clause

PostgreSQL's LIKE operator doesn't support [charlist], however SIMILAR TO does.

check HERE for a comprehensive list across DBs

Like @a_horse commented, you would have to use the regular expression operator ~ to use bracket expressions.
But there's more. I suggest:

SELECT *
FROM   tbl
WHERE  value ~ '^00[^0]'

^ ... match at start of string (your original expression could match at any position).
[^0] ... a bracket expression (character class) matching any character that is not 0.

Or better, yet:

SELECT *
FROM   tbl
WHERE  value LIKE '00%'       -- starting with '00'
AND    value NOT LIKE '000%'  -- third character is not '0'

Why? LIKE is not as powerful, but typically faster than regular expressions. It's probably substantially faster to narrow down the set of candidates with a cheap LIKE expression.

Generally, you would use NOT LIKE '__0', but since we already establish LIKE '00%' in the other predicate, we can use the narrower (cheaper) pattern NOT LIKE '000'.

Postgres can use a simple btree index for the left-anchored expressions value LIKE '00%' (important for big tables), while that might not work for a more complex regular expression. The latest version of Postgres can use indexes for simple regular expressions, so it might work for this example. Details:

• Difference between LIKE and ~ in Postgres