I think he's on the right track, but I suspect the O2 sensor is shorting to ground. This can cause the ECM to jump into self test mode after start up, and kill the pip signal. It would likely be in the wiring connectors, rather than a bare spot in the wires. F-150's, for example, have a harness connector to the O2 sensors below the battery, and it is common for it to corrode over time due to overflow expelled from a hot battery, and create the problem you described. Check your codes, and look for a code 18.
Take care,
~Chris
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