One chain we've discovered that works (at least the place we went to) is California Pizza Kitchen. We ordered the vegetarian pizza, but went cheeseless and subbed carrots, bean sprouts, and regular tomatoes for the oily eggplant, corn, and sun-dried tomatoes. I shudder to think all those times we ordered the regular veggie pizza, and figured that going cheeseless was good enough. We didn't know about the oil on the different veggies, but didn't care enough to ask either. (We found out what was possible by talking to the manager first.)
Sushi places sometimes have a nice selection of veggie sushi. Just skip the avocado and ask for low sodium soy sauce. (I've also skipped the soy sauce before and touched a bit of the wasabi paste to my sushi -- that works just fine.)
Local places (for those in the DC area) --
Zaytinya -- on the phone, they claimed that they could accommodate us, but then we show up and they say everything has olive oil. We kvetched a bit (we had needed a reservation of quite a few days in advance and we were shooting for a play ... we just couldn't go out and wander about Chinatown looking for food). We managed to get the cauliflower and mushroom dishes without the oil or nuts, plus salad with wine vinegar on the side. Turned out nicely.
We have another play coming up next weekend and are going to try Rasika. (We've gone before but not obsessing about oil.) They have said they will accommodate us, but we have to give two days notice I believe it is. Not a problem, since one has to make reservations about a month in advance anyway. I hope it works. They make everyone's list as one of the best Indian restaurants in the country (heck, one of the best restaurants, period).
Indique (Indian, next to Cleveland Park metro) -- we talked to the chef and got steam sauteed veggies with Indian spices inside a roti (the only Indian bread that works). It was good. Advance notice required.
In the Georgetown/ Kennedy Center area --
Notti Bianche -- give advance notice. We were given a custom risotto which had others looking on in envy. We were told that the next time the chef could devise something else.
Fino -- only one thing possible; gnocchi with red sauce (which they can make without oil)
Taj India -- had several things that they could make without oil. We couldn't have the rice because they put oil in that
but subbed in something else.
It is hard to find Italian. But our neighborhood place (Umberto's in Cabin John shopping center), has several dishes and they do not seem put out at all with doing no oil. It is now our "go to" place when we haven't planned far in advance and don't feel like veggie sushi. Maybe because Umberto's is actually run by folks from El Salvador? Anyway, if you go, split an entre and a veggie (be sure to specify no oil or cheese for both); it will be more than enough food. The house chianti is good (and inexpensive, too).
A final question -- the hummus and veggie sandwich at Cosi is 7 grams of fat, 59 calories of fat out of 397 (with 532 mg of sodium). Is this acceptable, a treat to be consumed rarely, or actually pretty good? Cosi's are all over, so it would give us a nice option.