Not a word from their mouth can be trusted;
their heart is filled with destruction.
Their throat is an open grave;
with their tongue they speak deceit.
This passage is referring to sinners who don't love or live in God. So i suggest that the line "Don't let your mouth become an old grave" should be changed to "Don't let your mouth become an open grave." and if you listen to the song, it actually sounds like he's saying "open" rather than "old."
Well, my take on this situation is... I do not think John Maus is interested at all in "sinners who don't love or live in God" or in considering this concept a viable way to view the world. Interpreting JM is not an easy task, and I am one brain, but based off his beautiful catalogue and distinct existential, "beyond good and evil" approach, when biblical terminology is used JM is not holding the view as a believer in the dogma enforced by the religious corporations that have taken possession of said texts.
Fortunately, the Old Testament in particular, is a beautiful, complex compilation that folk from all corners may embrace for their subjective edification.
Even though Don't Worship the Devil doesn't elaborate much lyric wise, you can definitely draw conclusions that are not related to any dogmatic, collective mentality. JM (and Ariel Pink for that matter) writes, sings, performs from a place very opposing to the conventions of pop culture (anywhere from music to organized religion). And that is a key to JM's music and why it is so affecting, the paradox of religious/pop melodies, drum machine, synth organ, fused with a deep current of chaotic, sensual, amoral, counter-moral... (etc) psychology.
"Don't worship the devil", the line in the song, is not uttered from a literal, traditional, dogmatic perspective, but from what the devil represents in our psychology as a symbol, first how it's kind of hilarious people's fear of this devil and demons (not to say spiritual forces do not exist, only that spiritual forces do not adhere to ideas proposed by a specific religion) and this is often self-inflicted, second as the vices and detriments to our personal being and community and that to engage in these destructive behaviors can undermine our character and relationships. To go so far as calling one a sinner and the other a saint is impossible without including that the saint is also a sinner, and the sinner at times a saint.
"Don't let your mouth become an open grave" - thanks for providing the lovely reference from Psalms. It is difficult to say much and be conclusive.