I would argue Yes.

Lay on Hands is a supernatural touch attack, meaning that it follows the rules of “weapon-like spells” found in Complete Arcane (yes, even though it is not a spell). These rules do not cover Smite Evil specifically, but generally clarify that touch-attack effects are effectively weapons in all ways but name – that is, if something enhances “weapon attacks” or “weapon damage” (e.g. Inspire Courage), it won’t work, but otherwise the touch-attack effect works like a weapon. Sneak Attack, for example, is explicitly called out as working. Rules Compendium clarifies further by saying that someone who has such a touch-attack effect is considered armed.

So in general, any touch attack that deals damage should be treated as a weapon in all but name. You make attacks with it, and things that improve attacks affect it. This should include Smite Evil. When you take your standard action to use Lay on Hands, you make an attack, which you may choose, at your option, to modify with Smite Evil by expending one of your daily uses of that ability.

Now, Smite Evil does say “normal attack.” What does that mean? No one knows. It’s not a defined game term and it’s unclear what counts as normal and what does not. Ruut’s argument that it means the Attack Action, which is used as a standard action, definitely seems wrong: not only is “normal attack” not the term they use for that (for that they say Attack Action), it also has pervasive impact on the paladin in general, making Smite Evil completely useless, rather than mostly useless. Jack’s argument that touch attacks are not “normal” attacks is more tenable – the rules, awkwardly enough, don’t have a good way to specify “not-touch attacks,” but ultimately this seems extremely flawed to me as well: does this mean a paladin cannot use Smite Evil in conjunction with a brilliant weapon? I don’t think so.

Ultimately, the paladin is a very weak class, and both Smite Evil and Lay on Hands are very weak abilities. The rules are ambiguous, but it is a very minor advantage to allow it, for a class that desperately needs every advantage it can get.

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.