I have had good luck not getting sick here. There are lots of places there can be cross contamination but as long as I don't eat anything towards the end of the counter closest to them (because they drag flour tortillas down the line and I've noticed them dipping into the bins of ingredients), I seem to be all right.
Chipotle does not have a gluten free menu but instead has a list of gluten-containing ingredients to avoid. Cross-contamination is a concern, however, so let the person taking your order know that you are gluten free and if you need them to take the proper precautions.
From their website: "If you avoid gluten, don't eat our flour tortillas. If you are highly sensitive to gluten and would like us to change our gloves, just let us know at the start of your order. You should be aware items containing corn, including corn tortillas (chips, soft corn, and crispy corn tortillas) and corn salsa, may have a small amount of gluten from potentially co-mingling with gluten-containing grains in the field."
Love this place. Bowls are easy to make GF. Love the corn soft tacos, but make sure you get someone behind the counter that is familiar with gluten allergies to avoid cross-contamination on the tortilla warmer. Also avoid the lettuce, as the gloved hands that made the last wheat flour tortilla were probably just in there.
This location changed gloves when I asked them to. No reaction. You would want cheese from the back because they use their gloves hands to serve the cheese.