I would like to layer multiple guitar lines for a section of a song that I'm working on. What are some harmonic variations that I would be able to use?
Usually tones from the chords you are plying over will harmonise well together; so as InternalConspriacy has mentioned 3rds/4ths/5ths will all sound good and interesting harmonised together; how you use these intervals will effect how the background chords sound as they change (by emphasising or blurring the change), this will depend on which chord you move to and which tones your are using. Mostly you will have to play it by ear.
You could also experiment with other intervals 2(9)/6(13)/7 will also sound good if creatively used, though the 2 and the 9 will have less of a prominent effect than the primary intervals.
Always listen to what the chords underneath are doing and to what type of chords they are; I tend to thing of triads/R5 chords as a blank canvass; whereas extended chords require a little more harmonic thought (eg: you might not want to play the 9 over a chord containing a flat 9).
I think the key is to get your first line down; and then experiment.
At risk of going very non-technical, have you got guitarists whose sound and harmonies you really like? Say for example you like the way Judas Priest used to layer guitar harmonies -it is relatively straightforward to describe what they did. Similarly for Lynyrd Skynyrd, or Yngwie Malmsteen or Al di Meola.
That could steer you towards making harmonic styles you already know you like, however it won't necessarily help you develop new ways to think about harmonies, which is one of the good things about the answers @DRL and @InternalConspiracy have written (I hadn't tried actively aiming for 9ths before - very interesting to play around with)
Honestly, I find it's always best to let the song tell you what it needs. While trying different 'ideas' is never a bad thing and can often yield very cool results, most often if I'm able to shut everything else out and just listen, I can hear what the song needs.
That being said, 3rds, 4ths, 5ths, sometimes 6ths, 7ths and the occasional 9 can be quite tasty...