Can a phone battery be used to power Arduino or Motors
It is very unlikely that a cellphone battery can power a motor shield, assuming motors will be actually operated off it. Motors typically draw large currents, and while modern Lithium-Ion batteries have fairly high current rating, they are designed to supply just enough for a small portable device. 

There are however very low-current motors available, such as pager motors, which can be run from a cell phone battery.

If the objective is only to run an Arduino board and conceivably a shield, but not operate a motor, things begin to look more positive. 

Success would depend on the specifics of the cell phone battery, but in general:


Many recent cell phones use Lithium-Ion batteries, with a nominal voltage of 3.7 Volts. This is insufficient for running any of the 5 Volt Arduino models, or any motor shield that I am aware of. This voltage can be used with some 3.3 Volt Arduino and clone boards: Not all 3.3 Volt Arduino boards support an input lower than 5 Volts, though.
The battery itself must allow power to be drawn from its contacts, without any special handling: Some cell phone batteries require a resistor of a specific value range connected between two of the tabs, to enable power delivery. Others may require (no specific example comes to mind) an electronic dialogue with the host cellphone, to enable charging / power delivery. 
Some older cellphone batteries supply a 5 or 5.5 Volt output (I wonder what battery chemistry they use). Such a battery will be better equipped to power a 5 Volt Arduino and similarly various 5 Volt shields.
Many cellphones contain a battery management circuit, used for controlled charging, battery monitoring and power delivery management. Bypassing this is not a good idea: Some types of battery will be damaged without battery management, or may no longer hold charge if drained beyond a point. 


How this can be done: 


Identify the positive and terminals of the battery (many have more than 2 terminals) and measure the voltage with a voltmeter. No reading = this battery isn't going to work.
Connect the above-mentioned 3.7 Volt compatible Arduino and the shield(s) to the two identified battery terminals.
Make sure the battery is recharged before it gets fully depleted. 




If the devices to be operated do not support 3.x Volt supplies, a boost regulator circuit can be used to boost the battery voltage to 5 Volts, with 75 to 90% efficiency. 

This will work, but the current delivery capability of the boost regulator will be lower than that of the battery alone.