O feitiço Porta da Dimensão ativaria ou contornaria o feitiço da Prisão Mental?

What it means to "be moved" is not defined in the rules so it falls to the Standard English meaning; for me, teleporting would count as "being moved" pelo feitiço and so will not bypass mental prison's dano

Note that "to move" is a rules-defined term meaning "to expend movement"^{2, 3} but using this to equate "to be moved" with "to be forced to expend movement" is probably not a good idea, and isn't supported in the rules anywhere. Thus we fall back onto the standard English meaning of "to be moved".

There is further support that "to be moved" uses the standard English meaning as shown in the "Ataques de oportunidade" section of the PHB:

You also don't provoke an opportunity attack when you teleport or when someone or something moves you without using your movement, action, or reaction.

This shows that someone/something pode move you without expending your movement/action/reaction which points towards "to be moved" as using the standard English meaning.

I personally would say that teleportation counts as being moved pelo feitiço and thus you are "being moved" which means that mental prison's damage will occur.

Adicionalmente a dictionary definition of "move" (from lexico) defines it as:

1 Go in a specified direction or manner; change position.

When one teleports one has certainly changed position, and thus they have moved. It's up to a GM to decide whether something (the caster or the spell) has actually moved them. But in my opinion if something moves, algo must have moved it. So regardless of how you've exited the area you have been moved out of it or moved through it and thus will take the damage.

Outra opção para porta de dimensão é como se segue:

[...] You can bring along objects as long as their weight doesn't exceed what you can carry. You can also bring one willing creature of your size or smaller who is carrying gear up to its carrying capacity. The creature must be within 5 feet of you when you cast this spell [...]

In this situation I would say the spell is ainda moving the chosen creature and so it will also not circumvent mental prison's damage. In fact, in this situation you could even say the caster is moving the chosen creature as well, so there's even more of a case for the creature having "been moved".

_{2: "Você pode se teletransportar para perto de uma criatura da qual tem medo?
3: "Does teleportation count as movement for a Tabaxi's Feline Agility trait?"}

Dimension Door would allow exit from the Mental Prison without harm.

Mental Prison states

[...] the target takes 5d10 psychic damage, and you make the area immediately around the target's space appear dangerous to it in some way. You might cause the target to perceive itself as being surrounded by fire, floating razors, or hideous maws filled with dripping teeth. [...]

[...] If the target is moved out of the illusion, makes a melee attack through it, or reaches any part of its body through it,[...]

o que significa que the target would only perceive that immediate area as dangerous. If the spell had not used the word 'immediately', then it would be clear that everything outside the target's space would 'appear dangerous to it'. However, since that word was used in conjunction with 'to it', then it can be reasonably argued that everything beyond the immediate area não appear dangerous to the target.

Further, of the three examples for being harmed after the illusion is up, 2 clearly state interacting with said illusion as the source of harm (by use of the word 'through'), leaving the third example, being 'moved'.

The question is, does the use of Dimension Door count as being moved?

Dimension Door states

You teleport yourself from your current location to any other spot within range. You arrive at exactly the spot desired. It can be a place you can see, one you can visualize, or one you can describe by stating distance and direction, such as "200 feet straight downward" or "upward to the northwest at a 45-degree angle, 300 feet."

So, Dimension Door allows you to disappear from one point, and appear in another, without even knowing anything about the destination (so being unable to see beyond your space as a result of Mental Prison would not prevent the use of this spell) by way of teleportation.

If we look at the rules on Opportunity Attacks:

You also don't provoke an opportunity attack when you teleport or when someone or something moves you without using your movement, action, or reaction.

we see that being teleported is explicitly distinct from being moved. So using Dimension Door would not count as being 'moved', nor would it cause any part of the target to reach 'through' the illusion.

This means that Dimension Door would allow the target to exit the illusion of Mental Prison without harm.

There are a couple of things that describe the victim of a Mental Prison.

Restraint and vision limitations

Whatever form the illusion takes, the target can’t see or hear anything beyond it and is restrained for the spell’s duration.

Restraint is not an issue for 5th edition's version of Dimension Door as the spell is verbal only. So unlike spells with somatic component there is no grey area in using Dimension Door.

For further details on the restraint condition and spellcasters see the following. Again the fact the Dimension Door is verbal only makes this a non-issue.

What Conditions prevent spell casting

Nor is the lack of vision beyond the illusion an issue.

Página 233 PHB

It can be a place you can see, one you can visualize, or one you can describe by stating distance and direction, such as “ 200 feet straight downward” or “upward to the northwest at a 45-degree angle, 300 feet.”

Leaving the Mental Prison

As a result the victim can leave the mental prison.

Page 161 XGtE

If the target is moved out of the illusion, makes a melee attack through it, or reaches any part of its body through it, the target takes 10d10 psychic damage, and the spell ends.

While the "if the target is moved out" is a bit confusing as its plain meaning suggests something else is moving the target out of the prison. It suggests that despite it being described as an illusion it is fixed to a location within 60 feet of the caster. Since Dimension Door results in the "target being moved out of the mental prison" the condition is triggered and the target takes 10d10 psychic damage.

As a side note the fact one took damage even though they teleported out would be a signature tell that the victim was subjected to the mental prison spell.