I have a Snow Peak 700. I never had the problem of burning lips because the cup cools down almost instantaneously coming off the stove/fire. However, the liquid inside did cool down fairly quickly. So I bought a GSI plastic cup with neoprene on it at REI, the one with a sippy lid. Its perfect. It fits inside the Ti pot along with my stove.

I am working on a pot cozy for the Peak 700. After a successful pot cozy I can eliminate the GSI cup for more space inside the pot to store more things.
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"To me, hammocking is relaxing, laying, swaying. A steady slow morphine drip without the risk of renal failure." - Dale Gribbel