2003 GMC Envoy Will Not Start Unless I Pull 3 40 amp fuses and then reconnect them?

I would have to say that the starter solenoid is weak (bad),so by you removing the fuses it allows the solenoid to reset , when you turn the key to start the engine the solenoid kicks in ,but is to weak to turn the engine over ,so it's causing the starter solenoid to hang -up until you turn the key off or until you pull the fuses. One more thing you should check is the starter relay ,it could be sticking as well.

You've definitely got a short circuit in the system somewhere. The fuse is doing exactly what it supposed to do, protecting the wiring/components from catching fire basically. We've had several Sierras in the shop for the same problem, the culprit was a bad starter every time. The electrical components on the starter fail and provide "excessive" resistance in the wiring causing the fuse to blow. In regard to diagnosis I would remove the starter and have it tested at a local auto parts store at no cost. You can replace just the ignition pack on those starters but I don't recommend it, for another $60 or so you can replace the entire unit. This has been the issue with every blown ignition fuse we've seen on late model GM trucks, yours might be something off the wall but I doubt it. Hope this helps! Good luck.