Fire risk is project dependent. It is impossible to explain the risks unless we see schematics, bills of material and photos of your actual devices and wiring.
An Arduino, powered by a proper cell phone charger or USB is a very low risk, especially if it is a legitimate Arduino with their normal fusing. Not all clone boards are fused so this should be verified if you are concerned.
The source of heating in electrical circuits is current. The higher the current, regardless of voltage, the more potential you have for heating. You can have steady state conditions that cause overheating, these are design errors such as a wire that is too small for the current passing through it or a resistor that is undersized.
Other types of overheating are related to failures causing uncontrolled or higher than normal current flow. This results in components or wire overheating during or after the failure. This is normally where fuses can provide protection, they should cut off the excessive current before things ignite.