Can an observer see that a creature is under the effect of the Mage Armor spell?
The closest we get to a description is:
You touch a willing creature who isn’t wearing armor, and a protective magical force surrounds it until the spell ends.
which tells us nothing about the appearance of this protective magical force.
Ask your DM for a ruling. Feel free to argue that it would be cooler if it were/were not visible, or that it would make more sense if it were/were not visible, or that the caster should be able to decide when casting, or whatever. There is no universal ruling.
nothing wrong with adding flavor to it. For example you could make you armor appear to encase your character like a guyver suit, witchblade armor or a Venom Symbiote activation. Another person might make it look like shimmering translucent armor almost like a celestial prortection. And still another might make it appear as a dome of protection barely visible unless it is "struck". Describing your spells is part of the fun I believe. After all since the Dr. Strange movie I see a lot of the Round arcane writing type of thing for shield spell now, that is obviously a very flavorful description of the spell.
Magical force is usually visible; mage armor remains ambiguous
Other spells/effects than mage armor deal with magical force:
- Armor of Agathys involves a protective magical force, but looking at the spell it is probably related to the cold counter-damage.
- Given that imprisonment's other effects all have visible manifestations, we could reasonably conclude that the sphere of magical force created by its "Burial" effect is also visible.
- Leomund's tiny hut "is opaque from the outside... but it is transparent from the inside."
- Otiluke's resilient sphere shimmers.
- Mage armor and Tenser's floating disk doesn't say what it looks like. (A disk might be a shape that would have to be visible to be discerned, or it might be simple enough that it need not be visible to be reasonably described as a disk.)
- Forcecage, shield, unseen servant, and wall of force are all invisible.
Additionally, PHB defines force damage type as "pure magical energy focused into a damaging form." The following spells/effects, then, also are related to/composed of magical energy:
- Banishing smite states that the imbued weapon "crackles with force" as it deals its extra force damage. This sounds like a subtle but noticeable effect (moreso to observers than the smitten).
- Bigby's hand is composed of "shimmering, translucent force," which is capable of dealing force damage as a possible effect.
- Disintegrate manifests as "a thin, green ray."
- Eldritch blast is "a beam of crackling energy."
- Magic missile creates "glowing darts."
- Given that Mordenkainen's sword is "sword-shaped," it is probably visible. (How could anyone tell it were sword-shaped otherwise?)
- Spiritual weapon is spectral.
- Most DMG magic items follow this trend of force damage being visibly inflicted.
- Beholders and beholder zombies deal force damage with their disintegration rays.
- Ghosts and other spectral beings are dealt force damage if they end their turn in a solid object, a visible effect.
- Poltergeists are invisible; it stands to reason their "Forceful Slam" is as well.
- Dimension door, etherealness, and teleport don't say what it looks like if they cause force damage.
Given that most effects dealing with magical force are visible, I would rule (and acknowledge that it is a ruling more than an answer) that mage armor is visible, probably as a shimmer or glow.