Can native Chinese speakers distinguish tones in songs? I already have a hard time distinguishing tones in normal speech. This gets significantly worse when I listen to Chinese songs, since the melody masks tones even further. 

Yes, I understand that context will help distinguish words from one another. But almost always I have to look up the lyrics to know exactly what they are singing about. 

Can native Chinese speakers tell tones apart in songs? How can I improve my tone-listening skills (in music)?
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.