The impression we got from reading the experience of people here that had such machines is that the machines rusted much faster than the ones that used electric heat.
I also have the impression (uncorroborated) that the safety requirements for equipment like that got much more expensive to implement and/or fix. One wouldn't want an open flame in an area that also might have oil/grease exposed when the transmission leaks etc.
Most of the (admittedly very) old washers with gas heaters tended to be top-loading horizontal axis with all the mechanism on one side of the drum and the flame under the outer drum, a lot of the machines didn't even have an arrangement to connect to a chimney, for example. I even saw pictures (taken by friends/family) showing some truly old washers from around 1900's or so that used wood or coal fire under the outer drum to heat the wash water in European installations, apparently the women in the building would get together in the basement on laundry day, some would prep the clothes, some would build a fire and get the water going, the washer was completely manual, users would let it wash for a while, remove the clothes with tongs and put the next load in, while other people would start rinsing the load etc.
On one hand, I admire the fact (if true) that a dozen or so residents could all get together and help each other, many hands light work and all. It's the kind of community one tends to aspire to. On the other hand, the cynic in me finds it hard to believe things were so "peaceful" and I tend to think people swallowed a lot of stuff they didn't want to just to keep the "peace", because I've seen buildings where folks can hardly get it together enough to run their loads thru the washers and dryers without causing trouble like leaving loads way past the time the machines stop, or other stuff (like say, leaving a pen inside a pocket, thus causing trouble for the next users).
Cheers,
-- Paulo.