You would need to compare the fuel costs in your area. Generally oil operates a bit cheaper but in some areas gas is cheaper but be careful when comparing fuel costs.

Figure out how much each fuel is per 1K BTU or Therm then compute the efficiency rate for each burner to determine how much energy you are getting for your dollar.

By example, the rule of thumb is that 1 Cu/ft of gas will produce 1100 btu however, if your burner is only operating at 75% your only getting 875btu of heat per cu/ft.

(I used 75% to simplify the math. Gas burners generally run much higher).

Also consider how you pay for fuel. In some areas they will put oil on a monthly payment plan, but here where I live now they will only deliver oil COD. For most ppl it is much easier to make a monthly payment than come up with $1500-$2k for a load of oil.

Gas is a much cleaner buring fuel so you don't have as much problems with soot or flyash buildup in the firebox or flues. In fact, most commercial boilers will operate on duel fuel, gas/oil. While oil is generally cheaper we often switch over and fire gas a few days each month to clean the firebox, thus reducing down time.

Oil generally requires more maintenance because you have pumps and nozzles that need adjustment or filters that need periodic replacing.

All heating systems should be professionally inspected and the operating system tweaked up at least annually, but for gas the procedures are much simpler and generally cost a bit less.

Where I lived in Ohio they do not permit installing oil but here in New England, where I live now, Oil is the most common heat source, although in the community where I live electric heat is the cheapest. The major downside with electric is that if you loose power you loose heat, whereas an oil or gas furnace can be run on a generator. But here again, in my community that is a non-issue. Our city has its own electrical generating company which began operating in 1898 and to this day they have only had two power outages that lasted more than 18hours.

My question wasn't worded well. What I meant is, are there any advanced steam boilers that run on gas. I consider tankless water heaters advanced over the tank version.

Have there been any sort of similar types of innovations in home steam boilers?

" I consider tankless water heaters advanced over the tank version."

Advanced, more sophisticated, yes. Economical, maybe not. Although they might be the right equipment in some cases, they are usually an expensive way to make hot water because of the up-front cost. You might even need a bigger gas service to install one.

You are going into the decision making process with a lot of incorrect assumptions. You need an expert HVAC person to come out and assess your home's energy supply and give you a recommendation as to how to best accomplish your needs for hot water and home heat. Picking one from column A and another from column B isn't the way to do it. The two decisions are tied together, and are dependent on a lot more variables than the very subjective description of "best" and "advanced".