Car doesn't want to stay running and overheats. We replaced all of the gaskets from the heads up. The heads were not warped. Any clue what the problem could be?

head could be warped or cracked bad rings or internal jacket damaged --do a compression test on engine and cooling system also pull plugs and check for abnormal burn

make sure you have the right cam timing. if its retarded it will overheat. did you use the proper cylinder head torque values and sequence? did you check if the cylinder head and or engine block is warped. start the car and it the upper radiator hose is getting stiff you are getting combustion pressure in the cooling system.

You say the head gaskets are fine, yet the compression is not.
How do you know the gaskets are not blown ?
Normally, when an engine runs hot, the gaskets expand and cannot contract, and the heads warp so they are not flat. You loose compression and usually have coolant in the cylinders.
You can also loose compression but not have coolant leaking into the block. It depends on what warped.

You will have to do a little bit of reading for this one : HowStuffWorks How to Tell If You Have Blown Head Gasket

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You have blown the head gasket. Also, you may have warped the head if it's aluminum which will happen due to ignoring the issue of overheating...which eventually caused the "blown head gasket". So, it sounds like you have 2 issues to resolve. 1) Why was it tending to overheat? Stuck thermostat, water pump issue or??? 2) Repairing the cylinder head or replacing it if warped badly, if your lucky it's just a blown head gasket. Allowing a overheating vehicle to be in service without solving that the overheating can lead to MANY other expensive problems.

If you overheat an engine with a cast iron block and an aluminium head, they will expand at different rates as they have different expansion coefficients. This will usually cause separation of the gasket and if you are losing coolant, it's a near certainty. The answer is a new gasket (pennies) plus a head skim at a local engineering shop, $100 or so (otherwise it won't seal again). You need a torque wrench and a workshop manual to check the torque settings and slackening/tightening order if doing it yourself, otherwise you'll cause further warping of the head. If the head isn't skimmed, it will go again after 50 miles or so.

"The car didn't overheat" you may have a blown head gasket, also if it sustained heat for a prolonged time , the head may have warped. Check coolant level.

waterpump? No movement of water/coolant, engine overheats. Engine overheats, warp heads. Next time you rip a head off, check gasket to see if head gasket failed or if there is a different cause to car overheating.

A new head gasket can't fixed warped heads. This engine must have seriously overheated at one time. Time for a new engine.

have you oil in the water? (radiater) if so you head gasket has gone and will nd replacing (big bucks) having a mix of water and oil gives you the sludge build up continuous over heating will certainly warp your head check your oil dipstick for a milky sludge instead of just oil