Does Elise continue to take damage during Rappel?

Yes Elise continues to take damage from debuffs applied to her during Rappel. She however does not take damage from Area Targeted effects during this time. It can therefore be used to 'dodge' many things, just not everything.

The wiki states, and I paraphrase;

'Untargetable' refers to champions in a state where targeted spells such as Rammus's Puncturing Taunt and LeBlanc's Sigil of Silence cannot be cast on the untargetable champion, and nontargeted spells such as Karthus's Requiemwill have no effect. Additionally, they cannot be targeted by auto attacks.

It also lists the description for Elise's Rappel skill as;

Rappel is a point targeted ability with two effects. If an enemy unit is targeted, Elise and her spiderlings will become untargetable for about 1 second and will move to the target's location. If Rappel is cast on the ground, Elise and her spiderlings climb into the air and become untargetable for up to 2 seconds. While in the air, Elise is granted vision of a large radius around herself, and may target an enemy unit within this radius with Rappel to descend upon them, moving to their location. If the 2 second duration expires, Elise will descend at her original position.

Nothing here mentions anything about single target debuffs that have already been applied, ongoing effects and the like, including but not limited to leashes such as Fiddlsticks' or Morgana's, or ongoing damage over time debuffs such as Malzahar's Visions or Summoner Ignite. Therefore it should be assumed they perform normally once applied.

sources:

http://leagueoflegends.wikia.com/wiki/Elise/Ability_Details

http://leagueoflegends.wikia.com/wiki/Untargetable