Truck starts but does not stay running

If this started right after the battery replacement I would start by cleaning the throttle body and checking that the idle air control isn't very dirty. What can happen is that dirt and carbon will build up in the throttle body over time. During normal operation the engine control module keeps opening the throttle and the IAC to keep the idle at the desired speed.
Now you disconnect the power and the ECM resets to the stock setting for a clean throttle body. That closes the throttle and IAC to the point that the engine cannot idle.
Not hard to clean using some throttle body cleaner and a rag. Disconnect the battery, remove the air duct from the throttle body and give it a good cleaning. Put it back together and start it up. The first couple tries it may take a bit of fiddling with the throttle before it will idle good. The easiest way is to get it started and take a drive so it can warm up. Then shut it off and restart.
https://www.2carpros.com/articles/throttle-actuator-service

Hi,

There could be a few things causing this. First, make sure there are no engine vacuum leaks. Here is a link that explains how that is done:

https://www.2carpros.com/articles/how-to-use-an-engine-vacuum-gauge

Next, the engine idle speed is controlled by what is called an idle air control valve. Here are the directions for testing that. The attached pic correlates with these directions.

2001 Toyota Truck Tundra Access Cab LTD 2WD V8-4.7L (2UZ-FE)
Air Control Valve
Vehicle Powertrain Management Fuel Delivery and Air Induction Idle Up Control Valve Testing and Inspection Component Tests and General Diagnostics Air Control Valve
AIR CONTROL VALVE
INSPECTION
1. TURN AIR CONDITIONING SWITCH OFF

pic 1

2. CHECK IDLE-UP
a. Start the engine and run it at idle.
b. Fully turn the steering wheel.
c. Check that the engine rotations decrease when the vacuum hose of the air control valve is pinched.
d. Check that the engine rotations increase when the hose is released.

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Start with these checks and let me know the results. Also, let me know if the check engine light is staying on when the engine is running.

Take care,
Joe