Does the burn from it double from nunu's iceblast, and when the slow duration ends does the double burn damage go down to normal burn damage right then? or does the burn damage stay double if they were slowed once during the whole duration?

I'm going to assume that if someones movement is no longer impared it will drop back down to 5%, but yes iceblast will activate it. This is likely something Riot will need to clear up in the text. As far building Lyandri's (sp?) I don't think I'd recommend it. After his ult ends I don't believe enemies are still slowed (if someone clears this up for me) so you'll only get the 5%, however at rank 5 I believe Iceblast will slow for the whole duration of the burn. He really only has 2 ways to proc the burn on champions.

Having a Nunu on the team and building a Lyandri as an AP will synergize very well together though. If you are looking for a strong AP item that works well on Nunu I'd look at the new AP jungle items.

In the end you can always test it out in a custom.

Liandry's Torment ticks 6 times over 3 seconds, once every half second starting from when the ability is applied. If they are slowed or otherwise movement-impaired during a tick, you will get the double damage; otherwise, you won't. Nunu, who pops some damage and then runs waiting for ice blast cooldown to be back up to do more damage, will be getting a good amount of damage out of Liandry's (especially once Ice Blast's slow lasts long enough to cover the entire duration).

^ Thoughts on AP nunu top?