What are the advantages / disadvantages for learning tones with Pinyin vs with Gwoyeu Romatzyh?
In general, I think too much importance is ascribed to which transcription system is used. Students' ability to distinguish, pronounce and remember tones depends on many factors and the way the tones are written is just a very small part of it. Even if there was a difference between systems, it would probably drown in other factors.

The question of tonal spelling (i.e. including the tonal information in the spelling of the word rather than using diacritics) has actually been researched experimentally. Scott McGinnis (1997)[1] compared two classes taught with either Hanyu Pinyin or Gwoyeu Romatzyh. They were tested on reading tasks and the accuracy of their tonal production was assessed by native speakers. The study found that the Gwoyeu Romatzyh group actually performed worse than the Hanyu Pinyin group.

Personally, I think the problem is that tones are often treated as something extra, even optional. Not by all, but by far too many. Some teachers don't grade an answer as wrong if the tone is wrong (or left out) but the initial and final are correct, or they don't deduct as many points for tonal errors as they do for spelling errors. Students often ask "do we have to remember the tones too?" and at one point an intermediate student asked me if I was serious when I said that he had to learn all the tones (his tones were terrible and it turned out to be because he had just ignored them).

So, changing transcription system to Gwoyeu Romatzyh will not help. The little evidence we have suggests that it won't and there's nothing else that suggests that it will. Yes, it's good to treat tone as an integral part of a word, but you can achieve that by consistent and persistent teaching or learning methods instead. Therefore, using Pinyin is the best option since it's by far the most well-spread and accessible system.

Going off on a tangent, I do think that it's worthwhile for students to learn more than one transcription system after they have already learnt their first. I think that this is good because it breaks the strong grip of orthography. Seeing the same sound written with different letters or symbols may lead to new insights into that sound. I have no research to support this, but it was like that for me and I know many others who have had similar experiences when learning a second transcription system, regardless of which one it is.

[1]: McGinnis, S. (1997). Tonal spelling versus diacritics for teaching pronunciation of Mandarin Chinese. The Modern Language Journal, 81(2), 228-236.