The key is the voltage on the LDOSENSE pin, pin 11. In normal operation, this pin should be at 0.5V. If the voltage is higher than 0.5V, then one of two conditions exist. Either something external to the TPS61130 is pulling the LDO to a higher voltage, or the load current on VDD is so small that the LDO cannot regulate.
If the voltage is lower than 0.5V on the LDOSENSE pin then either the pin is externally shorted to another trace, the voltage divider resistors are bad, or the chip itself is bad. To check for a bad chip, remove all power and allow all capacitors to discharge. Unsolder pin 10 to isolate it. Then check the resistance between pins 9 and 10. This should measure across the internal MOSFET pass transistor. If this reading is low ohms in both directions, then the internal MOSFET is shorted and the chip must be replaced.