I am playing in the G# Harmonic minor scale. I want to play a Dominant 7th chord, so would I base that chord off of of the D# minor or major scale? I am assuming it would be minor, but I am not totally sure, as I am still a theory noob.
G# harmonic minor is a derivative of the G# natural minor scale. If you are technically sticking to key this would make you in the key of B major. The V chord in B major is F# so you would use F# as your Dom 7. I may have misunderstood your question and not even answered it at all, let me know so I can try to make things more clear.
I do believe you have it right, sorta. G#m is in fact the relative of B Major. But due to my lack of wisdom, I can't verify if the rest of your response is correct, for if I could, I would've already known the answer. =(
EDIT: I realize it may seem as if I'm being a dick from my tone, I would like to assure you I am not, I'm just using 'proper English' as best I can, which often times makes one sound a bit conceded.
Hollowyears is right, F# Dominant 7 would be the Dom chord to use in G# Harmonic Minor.
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Hollowyears is right, F# Dominant 7 would be the Dom chord to use in G# Harmonic Minor.

This is absolutely INCORRECT.

You must treat the harmonic minor as its own scale, not as the natural minor. The major 7th degree of the scale changes all the chords involving that degree. It changes the i chord from a min7 to a min/maj7. It changes the III chord from a maj7 to a maj7+. It changes the iv chord from a min7 to a 7. It changes the VII chord from a 7 to a dim7.

Lets write out the G# harmonic minor scale : G#, A#, B, C#, D#, E, Fx
(*The Fx refers to F double sharp[F##])

Now we know that F# is not even a scale tone in G# harmonic minor, therefore F# would not be the dominant chord.

Lets write out the diatonic seventh chords for G# harmonic minor :
imaj7 - G#, B, D#, Fx
ii7(b5) - A#, C#, E, G#
IIImaj7+ - B, D#, Fx, A#
iv7 - C#, E, G#, B
V7 - D#, Fx, A#, C#
VImaj7 - E, G#, B, D#
viidim7 - Fx, A#, C#, E

From this we can gather two things:

1. The harmonic minor was made as an alteration to the harmonic minor, for harmonic reasons, hence the name. We know this because it produces different chords than a natural minor.
2. The important chords which have been changed were the v, and VII chords. A dominant fifth chord, and a diminished seventh chord, will give the strongest resolution to the tonic. The raised seventh gives us both a diminished seventh chord, as well as a dominant fifth chord, in a minor key.

Now, we know that the dominant chord in harmonic minor is the V7 chord. Thus, while playing in G#, we know that our dominant chord will be D#7.