This use of x works for me (I guess I didn't try it, because I would have chosen to post this version, not the other one):
Reduce[{x > -1, (1 - m + x) (E^x - 1/(1 + x)) > 0, 0 < m < 1}, {}, Reals]
Also this, which is closer to your code, Gianluca:
Reduce[{x > -1, (1 - m + x) (E^x - 1/(1 + x)) > 0, 0 < m < 1}, Reals]
It's the leaving off of the variable in the exponential that is the key. For instance, adding y as an explicit variable to your solution cause Reduce[] to fail:
Reduce[{x > -1, (1 - m + x) (E^x - 1/(1 + x)) > 0, 0 < m < 1} /.
x -> y - 1, y, Reals] /. y -> x + 1
