I listened to this several times and not entirely convinced that this is not done in a more simple manner. There are a few things at play here - multiple voices, delay/chorus (static comb filtering), and reverb.
The "up" specifically reminded me of something I did unintentionally on one of my very first 4-track recordings with a friend. We had set out to make chord progressions, lyrics, and record to tape an entire song in one sitting (c. 1989) using a Fostex 4-track. When we recorded the vocals, since the melody was more somber, I ended on a note an octave
below where he did, but then ramped that note up an octave to match him... turned out this "effect" sounded damn cool, so I did it two more times, and that was how it stayed. The "up" from that phrase reminds me a lot of this, and made me think that with delay/reverb that 4-track tape would sound similar (if I could ever find it).
I think the "up" is the result of the multiple voicings more than tape speed, which may be possible by duplicating a chorus track multiple times and using Melodyne (but never used Melodyne to pitch shift smoothly across several semi-tones... not sure how good it is with this). It seems that "effect" is a pitch shift in one voice (not all), similar to my experience long ago.
I still have a Tascam 424 Porta Studio under this desk, so will do a quick search to see if I can find that tape today. All the labels from tapes that old have fallen off, but that "effect" sounds so much like an intentional pitch shift by
the singer, rather than tape speed... reasons are that the "timing" matches, and it is not "all" of the voices. I think timing struck me more than anything, since this would be a clearer indicator, but never underestimate the human voice "all by itself."
Edit: This is the source I was listening to -
http://youtu.be/n4_TXuU9ars As I started playing other tapes in the background (the great "4-track hunt"), this comes across more and more as simply delay/reverb. The multiple voices is adding a dynamic to the track that requires multiple takes of the same phrase. As I have more background noise running, it is harder to detect any other effects on this phrasing.