Euchariste refers to the Christian sacrament of the Eucharist, also known as Holy Communion. It's a word with deep religious and symbolic meaning, making it quite rare and striking in a pop song.
In this song, Benjamin Biolay uses it in a very intense and almost sacrilegious way: "Je voudrais te dévorer comme un euchariste" (I would like to devour you like a eucharist). This line conveys an overwhelming, almost consuming desire for the person, elevating his passion to a spiritual, yet carnal, level. It's a powerful and memorable image that stands out.
Biolay sings to "Miss" as a lover he cannot do without, and the song keeps tying that love to death. In the first verse he swears "Qu'on va mourir sans s'oublier" (that we'll die without forgetting each other), with the two of them "Nus comme des vers" (stark naked) "Sur les sentiers de l'apocalypse" (on the paths of the apocalypse). She is "dans la peau" (under his skin), and dying is sad only because "C'est t'oublier toute entière" (it's forgetting you completely).
The second verse is the crash: "Je ne vois plus guère la lumière" (I hardly see the light anymore), "Je bois des 8.6" (I drink 8.6s, a cheap strong beer), "Je ne suis plus du tout fier" (I'm not proud at all anymore). By the last verse the hunger has turned strange, "Je voudrais te dévorer" (I'd like to devour you) like a communicant or "Un ébéniste affamé" (a starving cabinetmaker). The chorus admits the problem plainly: "Je ne peux pas non, me passer de toi" (I can't, no, do without you).