Do Linear regulators have a minimum current to maintain their output voltage?
According the the LM1117 datasheet, there is a typical minimum load current of 1.7mA (max 5mA) needed to maintain regulation. EDIT - this is for the adjustable version, I didn't notice you are using the fixed version, which has an internal resistor (which Steven mentions in his answer) to sink the minimum current required. So you will be okay, but the quiescent current (Typ. 5mA) is not great for a power conscious design, there are much better regulators out there if this is important. 

Usually a linear regulator will require a minimum load current to maintain regulation, the reason being a typical linear regulator cannot sink current, only source it. Since there are usually other control currents that flow to the output pin, then even with the pass element off if there is no load the output voltage can rise above it's desired voltage because there is "nowhere for the current to go".
If you study the schematic in the datasheet you should be able to see how this happens.