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. :D
There is a wordplay (if it is good or not you can 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) will mean 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) vs hassen (to hate). In second person they are equally pronounced.
And they repeat Du hast mich which wouldn't make sense (why everyone would think of to hate), not until they complete the sentence:
Du hast mich gefragt
As you already said, songs does not necessarily follow the grammatical rules. Same for poetries, etc. For the benefit of rhyme, metric or whatever the sentences can be rephrased in such a way that the are still understandable but are not perfect 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 previous paragraph) and as you can see on wiki the subjunctive of 3rd person singular (present active) is er/sie/es scheide.
In subjunctive the sentence is grammatical but, of course, there is a semantic error or at least the sentence is 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.