The correct subjonctive

There are 2 problems with your subjonctive try:
1. There must be a “que”
2. It is very odd to use “faire” here on the passive form

The correct subjonctive form would be :

Qu’il faille laver les assiettes

Use of “falloir” to mean “need to”

“Il faut que + subjonctif” means “need to”, so the correct sentence would be:

Il faut que les assiettes soient lavées

Difference between “falloir” and “devoir”

So here you have 2 ways to say the same thing :

Il faut que les assiettes soient lavées
Les assiettes doivent être lavées

The difference is very small, I think you can use one or the other in any context, without being misunderstood.
The first one slightly emphasizes the result (to be cleaned), and the second one slightly emphasizes the action (to clean).

Seul les assiettes doivent être lavées est correct.

Faille est le subjonctif du verbe falloir, qui s’emploie à la forme impersonnel (3ème du singulier).

Si on veut l’employer il faut dire : il faut laver les assiettes.