To be a dominant chord, it needs to first be a major triad; with a seventh, it needs to me a major triad with a minor seventh.
If we assume you are in C major, the iii (E-G-B) is a minor triad. With a seventh, it is a minor seventh (E-G-B-D). Neither of these will sound like a dominant.
If you were to raise the G to a G-sharp it would sound like a dominant, but it would "point" towards A minor, the relative minor. We would call it the V/vi. This is a fairly common occurence.
If you were instead to lower the B to a B-flat, it would sound like a diminished triad. This would lead us to F, and be called the iii0/IV.
EDIT: After @Todd Wilcox's comment of
"Can you elaborate on why, in the key of C major, the iii7 chord doesn’t sound and function like G/E? Or perhaps G/E no longer functions as a dominant chord with the altered bass note?"
I don't know for sure, but this is where my brain goes. In C major, the pitch B, or scale-degree 7, sounds very unstable. It wants to resolve upwards to C. But the interval of the perfect fifth has a stabilizing effect. I think (and this is just a hunch; I don't know how I'd prove it) that the addition of E in the bass provides a sense of stability. E sits a perfect fifth below B and gives it some context. That doesn't mean that you couldn't us this chord in place of a dominant. It just would never be as strong as a true dominant.