I record my acoustic guitar a couple times playing the same part, or the same part capo-ed up. I think that's called layering ... right? Sometimes I alter the mic position just to get some different sounds out of the guitar. My question is, can that cause phase problems in the same way using multiple mics can?

I believe that it can.  The closer the parts (tracks)  are to each other the more likely you are to have phase and comb filtering issues.

I use this technique quite often on acoustic guitar. I record the exact same part two times and pan them opposite for a fatter more spacious sound. Panning seems to keep that issue under control, at least to my ears and probably only because the sound is not in the same speaker where it will interact with the other track.
 
Any time you record the same musical part tow or more times, using the same mic position or even different ones, this issue can and will occur. The only question is, to what degree will it be apparent in the mix?

i run into this a lot, 
on both acoustics and electric,
when i use the same source.
 
the key is to change up enough of the sound chain, 
whether it's different guitars/amps, mics, positions,
to get the tones to play nice together.
 
but on some of my double tracking,
i play it so consistent, it almost sounds like a copy and paste of another track,
and THAT  is when i hear the phase.
typically, if i pan those on opposite sides, 
straight LCR mixing, it solves it.
 
sometimes i'll simply flip the phase on one of the tracks.
 
sometimes, i'll EQ the particular freq that stands out...
 
any or all of the above will be part of your next round of recording.
 

jrfrogers, this technique would more often be referred to as "double-tracking". "Layering" is usually used to describe sample-layering, which is similar except that it doesn't have the benefit of live-performance variation. But that's just how the terms are commonly used, not strict definitions.
 
Phase differences are actually important and desirable, as long as they're not static. It's the complex interplay between two close-but-not-identical tracks that adds fatness and interest. It's only when the two tracks are too close to being exactly the same that you get problems. So no, it's not the same as using two microphones on a speaker cabinet or a drum, because separate performances assure that the tracks are not identical.
 
I've run into phasing issues on double-tracked vocals. Back when AudioSnap first came out, I thought "great - now I can sync up those vocals perfectly!". So I did. Ugh. Comb-filtering defeated the thickening benefit of double-tracking, instead making the vocal sound thinner. I don't do that anymore.
 
What saves you with guitars is that it's impossible to play the same part exactly the same way twice in a row. That's why double-tracking works in the first place, whether you're talking guitars, vocals or handclaps. Natural human variation is the key.
 
You can further increase the necessary track-to-track variation a number of ways. Using different guitars is the best, but not everyone has that option. Other techniques for acoustic guitars: changing the microphone position and/or microphone, using a capo, different chord inversions, playing closer to the bridge, reversing up- and down-strokes, switching to a stiffer pick, moving to another spot in the room (or a different room), standing versus sitting. All of these things will assure differences that'll fatten the sound and avoid phase cancellations.
 
After tracking you can go even farther using EQ, compression and effects. One easy trick is to insert a graphic equalizer on each track and set each one to mirror the other. It's also easy with a parametric EQ, since you really only need a couple of filters in the upper-midrange. Modulated delays and choruses are another trick, but you have to keep it subtle.

It is highly unlikely multiple takes will cause phase cancellation problems.  Almost impossible actually.  The material would need near identical waveforms, not just similar sound.  I wouldn't worry about it.

TRUST YOUR EARS.
 

I was listening to a very intricate Clive Carroll fingerpicking piece earlier. It had obviously been double tracked and although he did an almost inhuman job of matching the two takes, the slight variations made this really awesome fat, organic chorus effect. I love it.

rumleymusic
It is highly unlikely multiple takes will cause phase cancellation problems with multiple takes.  Almost impossible actually.  The material would need near identical waveforms, not just similar sound.  I wouldn't worry about it.

Not impossible at all. I have not had this very often even with tight tracked guitars but certainly encountered this with vocals. I was recording this girl once and she double tracked so well it was almost perfect.
 
When I panned these vocals hard L and R of course nothing happened but as soon as I started moving those tracks towards each other I got severe and I mean serious comb filtering. The sound changed wildly and was very bad in parts.
 
Had to do some tricky polarity reversing in spots and time shifting too to keep it all OK in the end. So yes it can happen. I think it depends on the source and how tight the differences are. Stranger things have happened.

Thank you very much you guys - that helps a lot. I like the human differences in the double tracked guitar (not layered - thanks bitflipper) , and I do pan them L & R. My ears are not as trained as some of yours I can tell, so I thought I'd ask to see if this is something I should be listening for. I'm working the ears batsbrew.