-the boiler output is 60,000 BTU/hr with 6gpm and delta t 20f

OK, that makes sense, where delta T is the outgoing water temp - incoming water temp. Since 8.3 lbs/gallon * 6 gpm * 20F * 60 mins/hr = 60,000 BTU/hr.

-system mass is 100 Gallons

OK, I'll take that to mean equivalent mass of water. I.e. it includes the cast iron and other system components, in the sense that for the whole system, it takes 8.3 lbs/gallon * 100 gallon * 1.0 BTU/lb-F = 830 BTUs to raise the system temperature 1 degree F. [Cast iron has a heat a capacity of 0.110 BTU/lb-F, only 1/9 as much as water. So 830 lbs equivalent of water could mean 730 lbs of water and 900 lbs of cast iron, e.g.]

-boiler temperature min 160F max 180F

Is boiler temperature = outgoing water temperature? Regardless, I'm getting the sense you want to compute a length of time for a water temperature at some point in the system to rise from 160F to 180F while the boiler is firing.

Immaterial, if the emitter is still specified as emitting at 30,000 BTU/hr. The emitter behavior will actually depend on room temp and on incoming water temp, but if we are still to use the 30,000 BTU/hr figure specified in the OP, then that information has already gone into determining the 30,000 BTU/hr figure and we just need that result.

How long the boiler would take to supply the emitter with 180F water temperature?

OK, this question is well defined. If the emitters aren't emitting, and the boiler is dumping 60,000 BTUs/hr into the system, with no losses, then per the above it takes 830 BTUs to raise the system temp 1F. You want to raise it 20F, which takes 830 * 20 = 16,600 BTUs. 60,000 BTUs/hr = 1,000 BTUs/min, so the boiler would take 16.6 minutes to do that.

If the emitters are emitting at a constant 30,000 BTUs/hr, then the system is only heating up at a net rate of 30,000 BTUs/hr (60,000 BTUs/hr in from the boiler less 30,000 out from the emitters). In which case it would take twice as long for the system to see a 20F temperature rise, or 33.2 minutes.

Of course, that's an approximation, as the actual emitter behavior depends on the emitter incoming water temperature minus the room temperature. With an equation for the heat emission of the system as a function of system temperature, you could model the behavior of the system temperature over a firing cycle more accurately.

The upshot is that questions like this are a matter of energy accounting, tracking the BTU rates, and using the heat capacity to convert from BTUs to degrees F.

Cheers, Wayne