Can native Chinese speakers distinguish tones in songs?
No.

In songs, most tones disappear. The syllables are sung along the melody of music. We only tell the tone because we can catch what the whole word or sentence is.

In songs, usually the melody should be written to convey the tones of syllables (or syllables chosen to match the melody). Mismatch of melody and lyrics can result in misunderstandings. However, in pop music, not much attention has been paid to this problem.

An interesting paper (that this blog post pointed me to) claims that in Cantonese pop songs, the melody more closely follows the tones, while in Mandarin songs, the tones just get obliterated.