Do heal spells (Like Janna's Ult) proc Spell Vamp?

Short answer: NO

Spell vamp triggers on the DAMAGE dealed by spells.

None of the skills below deal any kind of damage, so spell vamp does not apply at any moment.

Spell vamp applies to everything that isn't classified as an auto-attack. This is most abilities, but also includes items (e.g. Hextech Gunblade) and summoner spells (e.g. Smite).

• Monsoon

(Active): Janna knocks surrounding enemies back and channels healing winds, restoring health to nearby allies each half second, for 4 seconds.

• Wish

(Active): Soraka fills her allies with hope, instantly restoring health to all allied champions regardless of where they are.

• Primal Surge

(Active): Nidalee heals an ally champion and grants them bonus attack speed for 7 seconds.

• Spell vamp

Spell vamp is a magic stat. The spell vamp works as a counterpart of life steal, but instead of regenerating health based on autoattacks, the regeneration is based on damage done by champion abilities.

No, they do not. Spell Vamp works specifically with "Damage." It does, however, work with Smite.