Starship is going to be fueled with cryogenic liquid methane. Before launch, the vehicle will spend some time fully loaded on the launchpad. Although methane is lighter than air at normal temperature, at -161.5°C (boiling point of liquid methane) it's 30% denser than air. It has to be heated another 30°C to become lighter again, which is not going to happen fast. Therefore methane can build up around the Starship in significant amounts even if refrigeration is considered, as sealing is never perfect and leakage can happen. Liquid methane expands 590 times. Considering its lower and upper explosion limits with air are 5.3-15.5%, ignition of the rocket could easily trigger a detonation. The danger may seem minuscule, but one cubic meter of gaseous air/methane (90/10) mixture is equivalent in explosion power to 190g of TNT. How does SpaceX plan to deal with this kind of explosion hazard around Starship?
Boil-off of methane is piped to a Flare Stack and burned off so it won't be vented unburned into the atmosphere. You can see a good picture of the flarestack on the right in the image of Starhopper fueling below:
(Image credit: NasaSpaceflight.com (bocachicagal)
Update Mar 4 2021: SpaceX no longer uses a flare stack. About six months ago they installed a methane recondenser, so the methane is turned back into liquid and reclaimed.
From an addendum to SpaceX environmental impact statement, submitted sometime in 2019-2020:
When the test vehicle is still connected to the ground systems, the methane would be transferred back to the methane tanks. If the vehicle is connected to the ground systems and tests are performed without engine ignition, the methane would be transferred to the flare. When the vehicle has been disconnected from the ground systems, such as during a static fire or test launch, methane would be vented to the atmosphere. Due to risks to personnel, SpaceX is unable to reconnect the vehicle to ground systems when methane remains on the vehicle. SpaceX is currently working to design a safe method to transfer the methane back to the tanks and reduce the amount of methane released.
As for when Starship is disconnected from the ground service equipment, it seems unlikely that there has been a separate re-condenser installed on-board the recent test vehicles.
Given that they are allowed to vent, and knowledgeable persons have tweeted that they indeed are doing so as recently as SN9, it seems some methane is still vented directly from the vehicle.