Background: Gamow-Teller (GT) transition strengths are key inputs for estimating weak reaction rates of importance for a wide variety of astrophysical applications. (n,p)-type charge-exchange reactions, such as the (t,He3) reaction used in this work, are commonly used for extracting the GT strength distribution in the β+ (electron-capture) direction. Such studies are important for testing theoretical models used to estimate weak rates for a large number of nuclei for simulations of astrophysical phenomena. Purpose: The Fe56(t,He3) reaction at 115 AMeV was measured in order to extract GT strengths for transitions to Mn56. The extracted strength distributions were compared with shell-model calculations in the pf-shell model space using the KB3G and GXPF1a interactions, and with calculations in the quasi-particle random-phase approximation (QRPA). Method: Differential cross sections and excitation-energy spectra for the Fe56(t,He3) reaction were determined by measuring the trajectories of He3 ejectiles through the S800 magnetic spectrograph and deducing their momenta. Contributions corresponding to GT transitions were isolated by using a multipole decomposition analysis. A well-established proportionality between GT strength and differential cross section at zero-linear-momentum transfer was utilized to convert extracted cross sections to GT strengths. Results and Conclusions: GT transition strengths from Fe56 to Mn56 were extracted up to an excitation energy of 10 MeV. Shell-model calculations with the GXPF1a interaction reproduced the observed GT strength distribution slightly better than calculations with the KB3G interaction. The calculated strength distribution in the QRPA did not reproduce the observed strength distribution. The new experimental data have an improved precision at low excitation energies compared to previous results obtained from an Fe56(n,p) experiment. Electron-capture rates based on the experimental and theoretical Gamow-Teller strengths were compared and deviations were included in an assessment of the validity of electron-capture rates based on theoretical models for nuclei in the pf shell.