I'm looking to use a switching supply from an old computer for a project. The project runs a motor that turns on and off based on a timer, powered by the switching supply. Do I run any risk running the switching supply while not under load (ie: when the motor isn't running?)
Generally, switching power supplies have a minimum percentage load to keep their output(s) in regulation. For example: a wall-wart type power supply rated for 12V may have a 16V output when measured with a DMM and no load. At very low loads, the power supply may also exceed its rated output ripple.
This problem, particularly for re-using computer power supplies, is generally solved by placing a power resistor that will draw the required minimum current across the affected supply rail. However, this decreases the system efficiency.
Additionally, depending on the configuration of a multi-output power supply's internal rails, the minimum power requirement may apply per-rail or to only a specific rail. For example, if the PC power supply generates 5V from the 12V isolated converter, externally loading the 5V rail would stabilize both the 5V and 12V rails. Check your datasheet, if available, or engage in trial-and-error testing for your specific supply.
You should have no problem using a switch mode PSU without a load.
However something to consider, a lot of modern switching supplies nowadays at low output power/load operate using "pulse" mode modulation. This means that below a certain output power, the power supply is not switched in a continuous PWM operation as they are normally, but the PWM is pulsed on and off at a much lower frequency than that of the PWM (below a few hundred Hz). This can present low-frequency ripple on the output, which for your application shouldn't be an issue.
It depends entirely on the individual supply. Modern supplies from reputable vendors generally have controllers that can maintain regulation under no load conditions though there can often be a low frequency ripple on the output which is difficult to filter. Older or low quality supplies can have significant overvoltage, significant undervoltage, massive ripple and other very bad behaviour.