(Edited to meet revised question: all of the systems I've left listed allow you to add an "objectively evil" aspect into the setting and deliberately frame the morality mechanics around it as needed.)

**Sorcerer**

Sorcerer is one of the strongest games to deal with morality in a pretty interesting fashion.  First, you define "humanity" as the moral aspects you want to deal with for this game story - a game based on samurai might deal with loyalty and duty, or a game about obsessed mystics might deal with human connections.

When character do things that affirm the value, there's a chance their Humanity score might go up  When they do things that break the value, there's a chance their Humanity score might go down.  The score itself doesn't dictate what you might do, or how your character IS, but rather is like the equivalent of moral hitpoints.

When you hit zero, you basically have two ways things can go (determined by the campaign you're running):

1. Character is lost (dead, corrupted, etc.)
2. Character is horribly changed for the worse - GM completely rewrites your stats and situation, reflecting your harrowing transformation/experience.

The GM is supposed to use intelligent judgment about what counts thematically as triggering a roll, the randomization sets up enough uncertainty to really create tension - you might be a really "good" person but get hardly any points, you might be a terrible person and never lose Humanity.  Generally the behavior pushes you one way or another, but it's never guaranteed.

**Tenra Bansho Zero**

TBZ has a Karma system which works in a very interesting way.  Your character has goals, relationships, and ideals known as Fates.  As you play pursuing these things, you use your Fates to create Karma points which you can use for bonus dice or character improvements.   Your Fates, however, are limited by a bit of math - you eventually have to commit more to one or two, and let go of the other ones - either giving them up entirely or changing your attitudes about them. 

It's a fun way of implementing the Buddhist idea of detaching and letting go of attachments as a character mechanic.

Now, characters who choose NOT to do so, become Asuras - or basically "obsessed demons".  This sets up this fascinating set of options for characters and morality in TBZ:

1. You stay stuck on a set of goals/relationships, and become an Asura
2. You stop spending Karma, and become a static character without much growth
3. You change your attitudes, views and let go, continuing your process of growth as a better person
4. You change your attitudes, views and let go, continuing your process of growth but with equally negative or worse views in it's place.

**Primetime Adventures**

Every protagonist in PTA has an "Issue".  These are typically personal issues as a person, character flaws, although social/power issues tend to bring your character into the same fields of conflict even though they're often more outwardly focused.

Scenes are set often aiming at one or more of the character's Issues, and good roleplaying is rewarded by the group with "Fanmail" (AKA, hero points), so you have good reason to wrestle with these things.   Each character gets a spotlight session to really try to resolve the Issue as a focus.

Although the rules are very light, PTA has been one of the best, consistently interesting games for me over the years.  Often the "Issue" players start with isn't exactly the real underlying focus for their character, but it serves as a great signpost and you dig up really meaty play this way.

**Burning Wheel, Riddle of Steel, Blade of the Iron Throne, Shadow of Yesterday**

I'm lumping all of these games together because they deal with Ideals based mechanics - each character has several ideals, relationships, or goals which directly give them bonus dice and XP or advancement points.  These can change over play, and ultimately what you see in terms of morality is a longer term aspect of how a character changes and grows (or falls) since the systems reward the pursuit of these ideals regardless of whether they're good or not.