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Opening doors and the hood was a thing on older Corvettes. Not needed at all anymore. As noted above, the C6 is so rigid that when you jack from the front jack puck, the rear of the car will also raise up. Your mechanic is working off of info that was relevant with C3's and maybe C4's.
Use the jack points shown above by Don-Vette and you will be fine. Always use jack stands AND wheel chocks so you car does not roll. Doors and hood can stay closed.
I had a 68 Vette and that thing flexed bad. As mentioned, the C6 is way more ridged.
You can use a combination of these jacking points depending on what and where you are working on under the car. The C-6 has a very rigid frame and very little flex. Just follow the diagram for jack and stand placement and you will be fine.
The C6 is actually a easy car to jack, if you only want one wheel up just jack from the puck locations. If you want to jack both front wheels you can use a jack in the center cross member, for most jacks to fit in the under the car in the center you will have to drive up on a couple two by sixes or use ramps. Pretty straight forward.