In my experience, you can use pg_dump without formatting to backup and restore databases between versions without any issues. This is when using pg_dump to simply generate a SQL file.
pg_dump mydb > db.sql
However, when using -Fc as an option
$ pg_dump -Fc mydb > db.dump
You can only restore the dump file with a version of PostgreSQL that's higher i.e. more recent. So if you create the dump file with PostgreSQL 8.1, you can only restore that same file with pg_dump from a 8.1 or more recent version of PostgreSQL.
There may be other elements that must exist on the target server such as plpgsql or postgis, if those same elements exist on the source server.
Here is the documentation from Postgres on pg_dump.
Here is the relative paragraph
Because pg_dump is used to transfer data to newer versions of
PostgreSQL, the output of pg_dump can be expected to load into
PostgreSQL server versions newer than pg_dump's version. pg_dump can
also dump from PostgreSQL servers older than its own version.
(Currently, servers back to version 7.0 are supported.) However,
pg_dump cannot dump from PostgreSQL servers newer than its own major
version; it will refuse to even try, rather than risk making an
invalid dump. Also, it is not guaranteed that pg_dump's output can be
loaded into a server of an older major version — not even if the dump
was taken from a server of that version. Loading a dump file into an
older server may require manual editing of the dump file to remove
syntax not understood by the older server.