Symbiotic recurrent novae are binary systems in which a massive white dwarf accretes hydrogenrich material from a red giant companion. Recurrent outbursts are triggered by thermonuclear runaways on the white dwarf surface. These binary systems are thought to be progenitors of type Ia supernovae. The 2006 outburst of the recurrent nova RS Ophiuchi has been observed at radio, infrared, optical and X-ray wavelengths. These observations have allowed to clearly identify the blast wave expanding into the red giant wind. The evolution of the nova remnant indicates that high-energy particles have been rapidly produced at the blast wave, most probably through the diffusive shock acceleration process. We have found that accelerated protons can have achieved in few days after outburst a maximum energy as high as a few TeV. The gamma-ray emission produced by pion decays from the nuclear interaction of relativistic protons with the thermal particles could have been detected with GLAST, if it was operating at that time. The prospects for the detection with GLAST of high-energy radiation from recurrent novae will be discussed. Such observations would allow to better understand the recurrent nova phenomenon and to further test and improve the theory of diffusive shock acceleration.