Yes, you remain invisible.

The text states:

When you are in an area of dim light or darkness, you can use your action to become invisible until you move or take an action or a reaction.

We can break it up to determine how it works.

When you are in an area of dim light or darkness, you can use your action to become invisible

This is the only condition that must be met for you to become invisible. If you are in dim light or darkness, and you can take an action, you can become invisible full stop.

until you move or take an action or a reaction.

These are the only conditions that cause you to lose invisibility. The invisibility ends when you either move, take an action, or take a reaction. Transitioning from dim light/darkness to full light is not moving, taking an action, or a reaction, so the invisibility remains.

If you're worried about this being overpowered, remember that in 5E invisibility is not that useful by itself. It doesn't actually make you any stealthier or harder to detect than normal. The only thing it does is give people disadvantage against you (which is powerful no doubt), and it allows you to hide in plain sight. However, hiding is an action, which according to the text you quoted removes the invisibility. Since your Warlock can't hide without losing his invisibility, he will still be detectable by everyone around him. All this lets your Warlock do is get disadvantage on attacks against him as long as he doesn't move or take actions/reactions.

Yes We Can...

On page 72 of Complete Arcane, the 'Specific Spell Requirements' section states:

A requirement based on a specific spell measures whether the character or creature in question is capable of producing the necessary effect, and as such, invocations and spell-like abilities that generate the relevant effect meet the requirements for specific spell knowledge.

Thus, with the lesser invocations Flee the Scene (Complete Arcane, p134)(replicating both dimension door and major image), Wall of Gloom (Complete Arcane, p129, 136), a gnome warlock would qualify partly via its own racial Spell-like abilities (dancing lights, ghost sound). This satisfies the racial and illusion spell requirements of Shadowcraft Mage (Races of Stone, p120-122).

To satisfy the 4th level Shadow subschool spell requirement, the warlock could take the Heighten Spell-like Ability (Complete Arcane, p80) feat, with Wall of Gloom as the targeted spell-like ability. It's eligible by 9th level, as it's 2 levels lower than half the warlock's caster level by then.

The warlock could become a Shadowcraft Mage at 10th level, and gets the Shadow Illusion ability by 12th level.

...Maybe

As to what would happen, if the DM agreed with the interpretation that the PC is casting the major image from Flee the Scene, then the PC would get all 3rd level Sorcerer/Wizard Evocations and Conjuration (Creation/Summoning) spells as at-will Spell-like abilities, at 40% efficacy at 12th level. This would increase to 60% reality by 14th level, at Shadowcraft Mage 5th, with the Powerful Shadow Magic ability.

Taking Heighten Spell-like Ability again at 12th for Flee the Scene would the warlock could mimic a 5th level Sorcerer/Wizard Evocation or Conjuration (Summoning/Creation) spell at 60% (eventually 80% at Shadowcraft Mage 5, level 14) reality 3/day.

As the Flee the Scene invocation would be a [Shadow] spell of 4th level, at that point, Wall of Gloom could be traded out for another invocation that mimics an illusion spell, such as Walk Unseen (Complete Arcane, p136). This should only be done if feat retraining (Player's Handbook II, p193) were allowed, in which case the Heighten Spell-like Ability feat should be retrained to focus on Flee the Scene at that time as well, saving a feat.