It also sounds as if the birthplace had moved, rather than the borders. If I continued to use the word "current" (rather than "modern", or some other phrase) would say Rumi's birthplace is not within Iran's current borders.
Usually, and to some extent this is a matter of style, the singular border is used for parts of the line that surrounds a country. We might refer to France's border (singular) with Spain, for example. If we wanted to discuss the entire line, we'd talk about France's borders.