I have a Cantus Firmus in D dorian mode that I am trying to write a 4 part counterpoint exercise with. In my textbook it says that since we have 4 notes per triad I need to double a note per triad to get a 4 voice voicing. It also says to double the degree 1, 4 and 5 of the scale I am in (it uses C major as an example which means the notes to double are C, F and G). However I get confused how this applies to modes? Since D dorian is part of the C major scale, do the same notes get doubled (C, F and G)? Or does being in dorian mean I need to use 1,4 and 5 of Dorian mode?
The 1, 4, and 5 being referred to are within the mode; that is, they are not 1, 4, and 5 of the corresponding major scale.
The general principle is not to double "tendency tones". Since you're dealing with triads, this means, in particular, the leading tone.
For example, in G - C cadence, you wouldn't double the B in the G chord.
This idea gets extended to say that you shouldn't double the third of a triad (the leading tone being the third of the V chord, for example). In C major, if one is dealing primarily with I, IV, and V triads, this means that 1, 4, and 5 (i.e. the roots or fifths of the triads) can always be doubled. However, 3, 6, and 7 are risky in that they could be the third of a I, IV, or V chord, respectively.
Doubling roots and fifths of major and minor triads is safe regardless the mode.
I'd like to add that the reason not to double "tendency tones" is that, because of the tendency, these "tend" to resolve to the same tone leading to parallel octaves. In C-major, (as an example), one doesn't double B as the tendency of B to move to C is felt rather strongly, especially as B is the third of the dominant. I know little about modes but the same ideas should apply. Tones in major or minor or modal tend to resolve by half-step if possible to another diatonic tone. I'd guess that (using the "white" notes from the natural hexachord) that moves of B to C, C to B, E to F, and F to E can have trouble with parallels.
The entire point is to keep the voiced independent. (In fact, in my stuff, my biggest problem is parallels between bass and melody. It's even worse when improvising.)