Because readers in PostgreSQL do not lock data, regardless of transaction isolation level, data read by one transaction can be overwritten by another concurrent transaction. In other words, if a row is returned by SELECT it doesn't mean that the row is still current at the instant it is returned (i.e., sometime after the current query began). The row might have been modified or deleted by an already-committed transaction that committed after the SELECT started. Even if the row is still valid "now", it could be changed or deleted before the current transaction does a commit or rollback.
Another way to think about it is that each transaction sees a snapshot of the database contents, and concurrently executing transactions might very well see different snapshots. So the whole concept of "now" is somewhat ill-defined anyway. This is not normally a big problem if the client applications are isolated from each other, but if the clients can communicate via channels outside the database then serious confusion might ensue.
To ensure the current validity of a row and protect it against concurrent updates one must use SELECT FOR UPDATE, SELECT FOR SHARE, or an appropriate LOCK TABLE statement. (SELECT FOR UPDATE and SELECT FOR SHARE lock just the returned rows against concurrent updates, while LOCK TABLE locks the whole table.) This should be taken into account when porting applications to PostgreSQL from other environments.
Global validity checks require extra thought under MVCC. For example, a banking application might wish to check that the sum of all credits in one table equals the sum of debits in another table, when both tables are being actively updated. Comparing the results of two successive SELECT sum(...) commands will not work reliably in Read Committed mode, since the second query will likely include the results of transactions not counted by the first. Doing the two sums in a single serializable transaction will give an accurate picture of only the effects of transactions that committed before the serializable transaction started — but one might legitimately wonder whether the answer is still relevant by the time it is delivered. If the serializable transaction itself applied some changes before trying to make the consistency check, the usefulness of the check becomes even more debatable, since now it includes some but not all post-transaction-start changes. In such cases a careful person might wish to lock all tables needed for the check, in order to get an indisputable picture of current reality. A SHARE mode (or higher) lock guarantees that there are no uncommitted changes in the locked table, other than those of the current transaction.
Note also that if one is relying on explicit locking to prevent concurrent changes, one should either use Read Committed mode, or in Serializable mode be careful to obtain locks before performing queries. A lock obtained by a serializable transaction guarantees that no other transactions modifying the table are still running, but if the snapshot seen by the transaction predates obtaining the lock, it might predate some now-committed changes in the table. A serializable transaction's snapshot is actually frozen at the start of its first query or data-modification command (SELECT, INSERT, UPDATE, or DELETE), so it is possible to obtain locks explicitly before the snapshot is frozen.