Consulta recursiva con sum en Postgres

Esta es una versión simplificada de la correcta de @a_horse responder (después de la discusión con OP en los comentarios).
Funciona con cualquier número (razonablemente finito) de niveles en la recursividad.

Precio total para el project_id dado

WITH RECURSIVE cte AS (
   SELECT project_id AS project_parent, project_value
   FROM   projects
   WHERE  project_id = 1 -- enter id of the base project here !

   UNION  ALL 
   SELECT p.project_id, p.project_value
   FROM   cte
   JOIN   projects p USING (project_parent)
)
SELECT sum(project_value) AS total_value
FROM   cte;

Precio total para todos los proyectos

WITH RECURSIVE cte AS (
   SELECT project_id, project_id AS project_parent, project_value
   FROM   projects
   WHERE  project_parent IS NULL  -- all base projects

   UNION  ALL 
   SELECT c.project_id, p.project_id, p.project_value
   FROM   cte c
   JOIN   projects p USING (project_parent)
   )
SELECT project_id, sum(project_value) AS total_value
FROM   cte
GROUP  BY 1
ORDER  BY 1;

db<>fiddle aquí (con el caso de prueba correcto)
_{Antiguo sqlfiddle}