Without a battery installed, the fan may run at high speed. It is also possible the laptop may run very slowly since the CPU may become throttled when the system doesn't see the battery thermal sensor. I don't recall whether the 2012 & earlier Apple laptops run slowly when the battery is removed (I think they ran fine though....it was only for diagnostic purposes anyway), but it has been a while since I worked on them.
Leaving the battery installed does pose a risk that the battery may start to swell which could deform or damage the Trackpad. However, if the battery is not seen by the system, then the battery likely is not getting any power to cause any swelling.
FYI, with the Retina model laptops the computer will run slowly with a CPU throttled to idle speeds. With the USB-C Intel Macs the performance can become so bad with the battery disconnected, that the laptop may never finish booting....it does vary by laptop. With the M-series Macs, they won't boot at all without the battery connected.
I only ever worked on one or two early model MBAirs so I don't recall how the battery is installed, but I know on the later 2011-2017 Airs the battery is easily disconnected and/or removed. You can try it & see how the system works.
Edit: Just make sure not to puncture any of the battery cells since that would be bad. If you do remove the battery, then make sure to protect the battery from accidental punctures. And do not dispose of the battery in the trash, you must properly recycle the battery. Personally I would just disconnect the battery & leave it installed in the laptop. I would wrap some electrical tape around the battery connector to insulate it & try tucking it in between the battery & Logic Board. This way the battery remains protected & you don't have to worry about battery disposal until you retire this laptop where you will need to have the laptop properly recycled.