My 2009 chevy suburaban temperature gauge is reading nothing. a warning came across the screen saying engine hot shutting off ac. the engine light also turned on. the coolant tank has fluid.

Trust me on this one: The only thing to do in a situation like this is Hook it Up to a Quality OVD2 Diagnostic Scanner, if you don't its just a guessing game as many things can cause this. Call a friend or family member who has a scanner and have it checked, or go to a local parr's store and pick up an inexpensive obd2 scanner- they plug into the diagnostic port under the dash on the driver side, this will tell you the source of the problem or at least point you in the right direction!

The fact that that you haven't lost coolant and your reservoir is full still is a good sign. Try these 2 things:

1.) Check the oil and see if it looks "milky" or a strange color that's not the typical color of engine oil. If it looks ok at first glance, Look carefully at the dipstick and see if the oil level appears to be over filled- you'll know because the dipstick will be showing that it's at a level above full. This indicates that water/coolant has gotten into your oil and you likely have a blown intake gasket, this is common on GM Trucks/SUVs with the 4.8L/5.3L/6.0L V8.

2.) When the engine is cold and hasn't been running for a while, pop the hood and then start it up, quickly look under the hood for any leaks- especially around the water pump S it may be the issue(and an easy fix), or perhaps a radiator/heater hose is busted/leaking(also an easy fix). Just look for any leaks, if you have a friend or family member around get them to help you look for leaks, use a bright flashlight(even if it's daytime) so you don't miss a small leak as they're not always as obvious to see as you'd think. IT's OK TO LET IT RUN FOR A FEW MINUTEA BUT ONLY 2-3 MINUTES AT MOST AS ANY MORE TIME WILL CAUSE THE ENGINE TO OVERHEAT AND THAT'LL MAKE MATTERS WORSE AND YOU COULD END UP WITH A HUGE REPAIR BILL FOR MAJOR ENGIE ISSUE(S) INSTEAD- 2-3 MINUTES IS OK BUT THE SHORTER THE BETTER, SO IF YOU SEE A LEAK FIND ITS SOURCE AND CUT THE ENGINE OFF IMMEDIATELY- DON'T LET IT RUN ANY LONGER FHAN ABSOLUTELY NECESSARY!

I just fixed this issue on a Tahoe like yours and it was the water pump, the last one I fixed was an intake gasket, more on all of this below.

Regarding the Gauge Issue: it sounds like your Temperature Sending Unit is Bad which is no big deal and not expensive- it simply communicates with your gauge to display the engine temperature.

The overheating could be a number of things but most likely is a blown intake gasket as the 4.8L/5.3L/6.0L V8 Engines In 1999-Up Full Size GM Trucks & SUV's are known for having this issue. I just fixed a friend's Tahoe with this issue, I've done it many times on these GM Vehicles- I just bought a Tahoe with the same issue.

Worst Case Scenario is a Head Gasket Blown(or both are) and your cylinder heads could be warped or cracked as we, but it's usually not so serious, especially if you didn't drive it or let it run very long at all after noticing the overheating issue.

Any good mechanic/technician can diagnose this easily by doing a few common tests like a compression test and a leak-down test of your cooling system, etc.

Don't let that information alarm you, it may not be a big deal at all and a simple fix like a bad water pump, replacing the temperature sending unit, intake gasket, or all of the above which isn't expensive or hard to do at home with some mechanical ability and hand tools, but don't attempt it if you don't have any experience doing things like this- if you have any mechanical experience and can follow instructions then your Tahoe and all vehicles with the 4.8/5.3/6.0 are way easier than most to do the various repairs I've mentioned above to!