I think one of the first design choices you have to make is: can preceded_by be NULL? The first item in the list has to do something with preceded_by. As a general rule making columns NOT NULL is prefereable; here I declared it NOT NULL, and the first row must reference itself as a result:

  nworks=# create table todo (todo_id int primary key, task text, preceded_by int not null references todo (todo_id));
  CREATE TABLE
  nworks=# insert into todo values(1, 'do homework', 1);
  INSERT 0 1
  nworks=# insert into todo values(2, 'clean bedroom', 1);
  INSERT 0 1
  nworks=# insert into todo values(3, 'walk the dog', 2);
  INSERT 0 1
  nworks=# insert into todo values(4, 'call mom', 3);
  INSERT 0 1
  nworks=# select * from todo;
   todo_id |     task      | preceded_by 
  ---------+---------------+-------------
         1 | do homework   |           1
         2 | clean bedroom |           1
         3 | walk the dog  |           2
         4 | call mom      |           3
  (4 rows)

  
  nworks=# update todo set preceded_by = 1 where todo_id = 4; -- call your mother first, it's more important
  UPDATE 1
  
  nworks=# select * from todo order by preceded_by;
   todo_id |     task      | preceded_by 
  ---------+---------------+-------------
         2 | clean bedroom |           1
         1 | do homework   |           1
         4 | call mom      |           1
         3 | walk the dog  |           2
  (4 rows)

I didn't do any operation to fix the sorting order once I updated item #4, so there's more work to do here. I don't think it's a practical solution.