It's a little bit of both. The coil needs the boiler to be some minimum temperature, or you'll not get the water passing through the coil hot enough as it flows through that water bath. But, some boilers don't work well if they cool off too much, so there is some minimum required on some. Many of the newest don't need to maintain a minimum temp and only fire it up when the system calls for heat. Most of the older ones can be damaged if the return water is too cold (a result of the supply being allowed to cool off too much). So, without knowing the make and model, it's hard to tell.
Most boiler aquastats have a high/low setting. You might be able to set the low to say something like 140 and the high to whatever is required when things need heat (maybe 180, or somewhere near there). They also often have a differential setting that gives it some flexibility about when it turns on. If yours is set up for an internal coil, it may not have that high/low/difference adjustment. You might be able to retrofit one to save some energy.