Link for Thanatopic Spell
Relevant text
A thanatopic spell pierces defenses and immunities that protect against death effects, negative levels, and energy drain, affecting the target as if the protective barrier did not exist.
Inflict X wounds and Harm are not any of the bolded effect types above, so they function as they usually do against undead.
Negative energy damage almost always heals undead (unless the spell specifically says it does not) regardless. As Snowblind notes, Thanatopic metamagic has no purpose being applied to such a spell.
A naive, literal reading of the rules might suggest that it does, since the rules don't actually come right out and say that negative energy never harms undead unless otherwise specificed. There is a FAQ that states that undead are healed by negative energy and harmed by positive energy, in the context of explaining how negative energy affinity works, but that doesn't quite spell it out. However, it's pretty clear that the spirit of the rules regarding undead and positive/negative energy is that positive energy heals the living and harms undead (unless a particular effect disallows one of those two, such as the spell Disrupt Undead which only describes an effect on undead specifically) and negative energy harms the living and heals undead, except for effects like the umbral dragon's breath, which specifically says it does not heal undead.
Doesn't the clerics channel energy class feature pretty much spell out how positive and negative energy works - or is it not ok to extrapolate from that? also the negative energy plane only damages living creatures. (an aside: positive energy plane makes no mention of how it affects undead).
I would treat an umbral dragons breath weapon as channel negative energy to harm, so it has no effect on undead.
Quite frankly, the spirit of Thanatopic Spell is that it hot-wires negative energy to be harmful to undead, with the added benefit of allowing status effects that undead are normally immune to to affect them. Considering that one of the example spells in the feat is vampiric touch, which does not have any of those descriptors, leads me to potentially believe that that spell was supposed to have negative energy attached to it.
Huh. Never noticed that. Yeah, Thanatopic spell says that it can let Vampiric Touch get around Death Ward... which doesn't actually protect against Vampiric Touch as far as I can see.
Weird. Worth a separate thread for FAQ purposes, I'd say.
As for the breath weapon, it just does nothing to undead.