FB auto-posting is supported natively (see Settings → Connections).
Until we support Twitter natively, you can auto-post from your profile RSS feed by using IFTTT.com (search IFTTT.com’s site for Letterboxd → Twitter recipes).Damiano Dentice commented
How does one make his letterboxd updates show on Facebook or Twitter? I've looked in the settings and I can't figure it out. :\