Context. Heartbeat stars are eccentric binaries that exhibit a characteristic shape of brightness changes close to the periastron passage, primarily caused by a variable tidal distortion of the components. Variable tidal potential can drive tidally excited oscillations (TEOs), which are usually gravity modes. Studies of heartbeat stars and TEOs open up new possibilities for probing the interiors of massive stars. There are only a few massive (masses of components ≳2 M_⊙) systems of this type that are known thus far. Aims. Using TESS data from the first 16 sectors, we searched for new massive heartbeat stars and TEOs using a sample of over 300 eccentric spectroscopic binaries. Methods. We analysed 2 min and 30 min cadence TESS data. Then we fitted Kumar’s analytical model to the light curves of stars showing heartbeats and performed a times-series analysis of the residuals searching for TEOs and periodic intrinsic variability. Results. We found 20 massive heartbeat systems, of which 7 exhibit TEOs. The TEOs occur at harmonics of orbital frequencies in the range between 3 and 36, with the median value equal to 9, which is lower than those in known Kepler systems with TEOs. The most massive system in this sample is the quadruple star HD 5980, a member of the Small Magellanic Cloud. With a total mass of ∼150 M_⊙ it is the most massive system showing a heartbeat. Six stars in the sample of the new heartbeat stars are eclipsing. A comparison of the parameters derived from fitting Kumar’s model and from light-curve modelling shows that Kumar’s model does not provide reliable parameters. In other words, the orbital parameters can be reliably derived from fitting heartbeat light curves only if the model includes all proximity effects. Finally, intrinsic pulsations of β Cep, SPB, δ Sct, and γ Dor-type were found in nine heartbeat systems. This opens an interesting possibility for studies of pulsation-binarity interaction and the co-existence of forced and self-excited oscillations.