Hi. I'm trying to translate a Scottish proverb into Irish Gaelic, but my own grasp of the language isn't sufficient for me to do it with any real confidence. Any help would be appreciated.
The Phrase is "Cha bhi suaimhneas aig eucoir no seasamh aig droch-bheairt". The English translation should be "Wrong will not rest, nor will ill-deeds stand".

Cha bhi suaimhneas aig eucoir no seasamh aig droch-bheairt

The (Ulster) Irish equivalent is:
Cha bhíonn suaimhneas ag éagóir ná seasamh ag drochbhearta

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Is fearr Gaeilg na Gaeltaċta ná Gaeilg ar biṫ eile
Agus is í Gaeilg Ġaoṫ Doḃair is binne

Even the dialect I prefer! Go raibh maith agat.

Hi, Folks. The OP is after some confirmation, if possible.

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Múinteoir Gaeilge - Irish Teacher
My "specialty" is Connemara Irish, particularly Cois Fhairrge dialect, but I can also speak Ulster and Munster Irish with native-level pronunciation.
Is fearr Gaeilge ḃriste ná Béarla cliste, cinnte, aċ i ḃfad níos fearr aríst í Gaeilge ḃinn ḃeo na nGaeltaċtaí.
Gaeilge Chonnacht (GC), go háraid Gaeilge Chois Fhairrge (GCF), Gaeilic Uladh (GU), Gaelainn na Mumhan (GM), agus Gaeilge an Chaighdeáin Oifigiúil (CO).

I ran it past someone who translated it as "Injustice has no peace and a bad action has no standing", which is generally fine, but I'm curious/concerned about the will not vs. has no distinction at play.

It's a matter of present tense vs. future tense (obviously) - will not have vs. has no. It appears to me that the phrase is in the habitual/continuous present tense (certainly the Irish version is) - but this can easily be changed to the future tense. It might not be in keeping with normal Gaelic proverbs to put it in the future tense though.

Modern Irish has a simple present (or "habitual") tense, as in cha bhíonn in the Ulster dialect or ní bhíonn in other dialects. Scottish Gaelic does not have a habitual tense anymore, so as Scooby noted they use either the future (cha bhi) or (in some cases) the present progressive (chan eil [mi] [a'/ag] ...) where the habitual tense would be used in Irish. Which is used can depend on the context or meaning, but using the future predominates, especially in things like proverbs and sayings.

Lughaidh explained what happened to the habitual in Gaelic at this thread:
http://irishlearner.awyr.com/phpBB3/viewtopic.php?f=29&t=1563

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I'm not a native (or entirely fluent) speaker, so be sure to wait for confirmations/corrections, especially for tattoos.

By the way, in Irish you can't use the future tense after "cha", you have to use the habitual present instead, so :
beidh mé (I'll be) becomes "cha bhím" (I'll not be).

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Is fearr Gaeilg na Gaeltaċta ná Gaeilg ar biṫ eile
Agus is í Gaeilg Ġaoṫ Doḃair is binne

So when translated into Ulster Irish it has to be present tense? And that being the case, this is in fact an faithful translation?

Yes, if you use the future tense after "cha" it's wrong anyway
Btw not whole Ulster uses "cha", it's common mainly in parts of Gaoth Dobhair, Cloghaneely and in areas in the east (Ros Goill etc). The more southern Donegal dialects use mainly ní, so in Ulster you can also have "ní bheidh" instead of "cha bhíonn".

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Is fearr Gaeilg na Gaeltaċta ná Gaeilg ar biṫ eile
Agus is í Gaeilg Ġaoṫ Doḃair is binne