The progenitor stars of several Type IIb supernovae (SNe) show indications of extended hydrogen envelopes. These envelopes might be the outcome of luminous energetic pre-explosion events, so-called precursor eruptions. We use the Palomar Transient Factory (PTF) pre-explosion observations of a sample of 27 nearby SNe IIb to look for such precursors during the final years prior to the SN explosion. No precursors are found when combining the observations in 15-day bins, and we calculate the absolute-magnitude-dependent upper limit on the precursor rate. At the 90% confidence level, SNe IIb have on average <0.86 precursors as bright as an absolute R-band magnitude of -14 in the final 3.5 years before the explosion and <0.56 events over the final year. In contrast, precursors among SNe IIn have a >~5 times higher rate. The kinetic energy required to unbind a low-mass stellar envelope is comparable to the radiated energy of a few-weeks-long precursor that would be detectable for the closest SNe in our sample. Therefore, mass ejections, if they are common in such SNe, are radiatively inefficient or have durations longer than months. Indeed, when using 60-day bins, a faint precursor candidate is detected prior to SN 2012cs (~2% false-alarm probability). We also report the detection of the progenitor of SN 2011dh that does not show detectable variability over the final two years before the explosion. The suggested progenitor of SN 2012P is still present, and hence is likely a compact star cluster or an unrelated object.