I'm having a new Trane XR-13 A/C installed for $3,416 (which includes a 10-10-10 year warranty). I wanted to upgrade my thermostat but the company wants to charge me $375 for a Trane 602 thermostat. This same model which Honeywell makes can be had for $79 at Home Depot (Honeywell Model RTH7400D).

If I had someone other than this company install a Honeywell thermostat from Home Depot would this void the Trane warranty?

I believe it. HVAC parts upcharges are notoriously huge. Check your contract closely. It should not void the warranty, but I've seen all sorts of funky clauses in warrantee's.

(Also, just so you know, thermostats come with very good instructions nowadays. Buy it and try to install it yourself before calling someone out to do it - which will probably cost you $50-$75)

If you want the Honeywell thermostat buy it install it yourself and save the one that you took off. If you have problems under the warranty take off the Honeywell thermostat and put back on the one you took off before you call for the warranty service.. Sounds like cheating but so does $375 for an up grade from the installer.