A dhampir antipaladin takes a standard action to use touch of corruption, but there's more to it than that

There are, in fact, several issues with a dhampir trying to use the antipaladin's supernatural ability touch of corruption to heal himself:

1. The touch doesn't mention that it can be used as a swift action at all. While similar to a paladin's supernatural ability lay on hands, the antipaladin's ability just completely omits that part. The dhampir antipaladin must take a standard action to use the supernatural ability touch of corruption and, as part of that action, touch himself just like he would anyone else. However, it's likely he won't want to. That's because…
2. The touch does not deal damage because of negative energy. Instead, the "fiendish flame" deals untyped magical damage. (I know! Not even fire damage!) In most cases, this is awesome as that's one of the game's best kinds of damage: hardly any creature reduces such damage, and such damage bypasses damage reduction and spell resistance. But even if the damage were due to negative energy, the damage still wouldn't heal a dhampir antipaladin because…
3. The touch specifically and only heals creatures with the type undead. While this makes the touch useful when an antipaladin wants to damage, for example, a construct with immunity to magic (like a flesh golem), the touch isn't useful when, for example, a dhampir antipaladin needs a few extra hp to see him through a fight and he's all out of potions of inflict light wounds; a dhampir's type is still humanoid despite him being healed by negative energy due to the weakness negative energy affinity.

(Just in case the d20PFSRD failed to mention that touch of corruption really is negative energy, I checked Paizo's own SRD and neither is touch of corruption mentioned there as being negative energy, nor is it said to be such in its original form on page 120 of the Advanced Player's Guide.)

That's a fair number of house rules a GM must make if the antipaladin's supernatural ability touch of corruption is to function the way I expect you want it to. For comparison, the FAQ says the paladin's supernatural ability lay on hands as uses positive energy, but there's no corresponding FAQ entry saying the touch of corruption uses negative energy. While I understand the desire for the antipaladin to be the equal and opposite of the paladin, that doesn't seem to hold true in this case

"Would it be unfair to allow a dhampir antipaladin to take a swift action to use touch of corruption to heal himself?"

Maybe. If my PC were a paladin (or another PC with lay on hands) in the campaign, and if the campaign weren't undead-heavy, making only the above change but otherwise playing everything else as written allows another player's dhampir antipaladin to heal himself and battle with greater efficacy most foes better than my PC paladin who, unless battling a rare undead foe, can only heal himself. So, yeah, I would see that as a little unfair. In fact, I'd maybe have my paladin steadily become more and more bitter and sullen until he finally sought out someone who could offer him a deal that would even things out between my PC and his ally, even if it meant that, technically, my PC wouldn't be, y'know, alive anymore… I mean, undeath sounds unappetizing, but unlife? That's just like immortality but with a slightly different flavor, right…? And all that juicy power…?

Anyway, absent another PC in the party with the supernatural ability lay on hands, making the above change to the supernatural ability touch of corruption probably really won't matter to anyone but the dhampir antipaladin, but a GM should be prepared to make similar changes to other classes if it does end up mattering, just in case. Seriously, there's no way for me to really know what anybody else's group thinks is fair.

Very few feats published in official supplements or Dragon touch lay on hands:

• Action Healing (Faiths of Eberron): spend an action point to lay on hands with 50% extra healing, lay on hands as a swift action, or lay on hands from up to 30 feet away. Even if we assume you’re using action points, action points are way more valuable than those effects.

• Azure Touch (Magic of Incarnum): Grants 1 point of essentia, and allows you to basically trade away essentia for the day to gain another use of lay on hands (or wholeness of body). Essentia is way more limited and valuable than uses of lay on hands, so that’s a pretty terrible trade—the bonus point of essentia is by far the more valuable thing from this feat.

• Serenity (Dragon Compendium): Use Wisdom instead of Charisma for all paladin abilities, lay on hands included. Fantastic feat, but because of divine grace, not because of lay on hands.

• Touch of Silver (Faiths of Eberron): Burn a use of turn undead to make lay on hands hurt lycanthropes and evil outsiders as if they were undead. Which is, again, a terrible trade.

In three out of four cases, these feats involve trading away some limited resource in order to get some minor effect: these are terrible feats. Serenity is good, but not particularly because of lay on hands.

Which kind of works out, because lay on hands is a very weak ability. It is a really bad idea to invest feats, which are very scarce and extremely valuable, into it.

I do not know of any third-party feats that significantly change this conclusion, but I am also generally unfamiliar with third-party material.