A Google search gave me some information which motivated me to hear the song again and I perceived an interesting fact which I’ve never recognized before. I should listen more carefully to songs in future.
There is a wordplay (if it's a good one or not, you have to decide on your own). The second to last repetition says:
Willst du bis zum Tod der Scheide
Sie lieben auch in schlechten Tagen
Willst du bis der Tod Euch scheidet,
Treu’ ihr sein für alle Tage
That means, they refer to the female genital! And I think that sie lieben (also sang once) refers to to make love rather than to love.
Note that this is the only time they omit the t. In the other parts before, they don’t elide the t. Also become aware of the fact that the whole song is based on another wordplay, namely haben (to have) versus hassen (to hate). In second person they are equally pronounced.
They repeat singing "Du hast mich" a couple of times, which doesn’t really make any sense and that's why everyone thinks of to hate in the first place, i.e. "Du hasst mich.". But once they complete the line, it does make sense:
Du hast mich gefragt
As you already said, songs do not necessarily follow grammatical rules. Same for poetry etc. For the benefit of rhyme, metric or whatever, the sentences can be rephrased in such a way that they are still understandable but are not 'correct' German (same for English).
Regarding the rules of German (and which was my first (wrong) interpretation):
Talking in indicative you are right. Then there would be a t at the end of the word. But if they leave out the t, the sentence would be phrased in subjunctive (for the reason given in the previous paragraph) and as you can see on wiki, the subjunctive of third person singular (present active) is er/sie/es scheide.
In subjunctive, the sentence is grammatical but, of course, there is a semantic error; or the sentence is at least semantically disputable.
A short overview of the song on wikipedia says that the part is Willst du bis zum Tod, der scheidet, sie lieben auch in schlechten Tagen?. I heard the part again and again but I can’t hear a t; so I insist that the t is elided and the wiki article is wrong (not least because there are enough Google hits which agree).
I hope I’m allowed to link Youtube videos here. Have you already listen to the English version of the song? That’s absolutely not ambiguous:
Will you ’til death be her rider
her lover too, to stay inside her
Also have a look on this overview.