Unfortunately there is no single answer. Various issues make this answer a bit complex, so - can I lay my compressor down - has to responded to in a number of ways. Here are the things to consider when you go about moving your air compressor.
After it's been is use for some time, any air compressor will start to build up gunk in the bottom of the compressor tank. This gunk is made up of dirt particles that have bypassed or slipped through the intake filter and have been driven into the tank along with the compressed air.
Even if the tank is drained religiously (and really, do you actually drain the tank after every use like it is supposed to?) some moisture remains in the compressor tank. This moisture mixes with the ingested dirt, creating a sludge.
If the compressor is laid down, the drain and accomanying sludge will migrate around inside the tank. If that sludge gets to your tank check valve - which is commonly located where the air line from the pump enters the compressor tank - it will foul it.
Symptoms of this happening can be a cotinuous air leak from the compressor unloader valve, or a buidup of back pressure increasing the load on the motor as the tank check valve itself becomes mired in this crud. If the compressor is left for some time, that crud will dry on the flapper or ball in the tank check valve, preventing it from moving and doing it's job.
The increase in back pressure may overload the motor, and the motor may hum, overheat, turn off on thermal overload... or the compressor may not start at all.
That same sludge can roll over the compressor tank drain too, possibly plugging it so that no air or water can escape the next time the tank is drained.
Portable air compressors can be purchased in an oil-lubricated, or oil-free configuration. In the former, the compressor needs to have oil added to a sump. When the compressor is running that oil splashes up onto the crank and piston from underneath, lubricating the parts.
If the compressor is an oil-free model, the compressor has "lube for life" from the factory, and it is not required to add oil and no splash lubrication of the compressor pump occurs.
After a splash or pump lubricated compressor has been in use for a while, the oil gets pretty dirty. That, by the way, is why oil used in compressors is detergent free. Since there is no oil filter on lower end air compressors, any debris in the oil must be allowed to sink to the bottom of the sump, to be drained away when the oil is changed. That is unlike car or truck engines. In those, detergents and additives are included in the lubricating oil to keep crud in suspension in the oil and to allow it to be filtered out by the oil filter as the oil circulates.
Just like you normally would not turn the car engine over and allow dirty oil to flow into areas it was not supposed to go, laying an oil lubricated compressor down could allow the lubricating oil, and any crud that had settled, to flow into areas in the compressor it is not supposed to go, contaminating those areas with grit etc.
Can I lay my compressor down to move it? Sure you can, just as long as you understand the ramifications and maintenance issues that my occur as a result. For me, no way. I'll tie it upright, for sure.