Normally if you just walk into a regular restaurant, you won’t have much luck in finding a vegetarian dish there.
Now a days there are vegetarian and vegan restaurants opened in Tehran and you can find them with a simple search on google maps. You can also get lucky in some high-end restaurants in major cities or to find street food like Falafel sandwich (slightly different from Lebanese Falafel) or Sambuseh (similar to Aloo Samosa of India).
But except when you are traveling in province of Gilan and especially city of Rasht (which is Food Heaven of Iran!!) your only chance of tasting a delicious vegetarian dish is at Iranian homes.
Not only most of the population were pastoralists and meat and dairy were always present in their food, but also eating meat used to indicate a higher social status, as it showed the ability to afford meat. (Same applies to consuming white rice and white bread, which used to be scarce and some elderly still believe that refined grains are healthier than the whole grain ones).