What if our business requirements are such that the data can only be added and modified in some tables, but not deleted?

One way to handle this will be to just revoke the DELETE rights on these tables from all the users (remember to also revoke DELETE from PUBLIC), but this can also be achieved using triggers because of reasons such as auditing or returning custom exception messages.

A generic cancel trigger can be written as follows:

CREATE OR REPLACE FUNCTION cancel_op() 
  RETURNS TRIGGER AS $$ 
BEGIN 
    IF TG_WHEN = 'AFTER' THEN 
        RAISE EXCEPTION 'YOU ARE NOT ALLOWED TO % ROWS IN %.%', 
          TG_OP, TG_TABLE_SCHEMA, TG_TABLE_NAMENAME; 
    END IF; 
    RAISE NOTICE '% ON ROWS IN %.% WON'T HAPPEN', 
          TG_OP, TG_TABLE_SCHEMA, TG_TABLE_NAMENAME; 
    RETURN NULL; 
END; 
$$ LANGUAGE plpgsql; 

The same trigger function can be used for both the BEFORE and AFTER triggers. If you use it as a BEFORE trigger, the operation is skipped with a message. However...