  The particle origin of dark matter (DM) is still one of the main puzzles in
modern physics. One of the most promising search strategy to detect DM at
laboratories is through the indirect search of cosmic particles that are
produced from DM annihilation in space. In particular, the flux of cosmic
positrons has been measured with high precision by the AMS-02 experiment
demonstrating that an excess above 10 GeV, with respect to the secondary
production, is present. We study in this paper the possible DM origin of the
positron excess finding the values of the DM mass $M$ and annihilation cross
section $\langle \sigma v \rangle$ that are needed to fit high-energy positron
data. In particular, we find that for DM annihilating into $b\bar{b}$ it is
required to have $M=43$ TeV and $\langle \sigma v \rangle = 10^{-21}$ cm$^3$/s
while for $\tau^+\tau^-$ $M=2$ TeV and $\langle \sigma v \rangle = 3\times
10^{-23}$ cm$^3$/s. If DM produce positrons, they are expected to generate
gamma rays from the center of the Milky Way and around dwarf galaxy satellites
of the Galaxy. We thus combine the values for the DM mass and annihilation
cross section obtained with the fit to AMS-02 positron data with the upper
limits derived with the non-detection of $\gamma$ rays with HESS in the
direction of the Galactic center and Fermi-LAT for the combined analysis of
dwarf galaxies. The main result of the paper is that only DM annihilating into
$\mu^+ \mu^-$ with a mass around 500 GeV and $\langle \sigma v \rangle =
4\times 10^{-24}$ cm$^3$/s can fit AMS-02 data and be compatible with the upper
limits found with $\gamma$ rays. As for the $\tau^+ \tau^-$ ($b\bar{b}$)
channel, DM can contribute at most at a few tens $\%$ (a few \%) level.
