The linear stability analysis of Oldroyd-B fluid for thermocapillary liquid layers is carried out. Results are presented for linear flow and return flow with Prandtl numbers of 0.02 and 100. Three kinds of instabilities are found: oblique wave, streamwise wave, and spanwise stationary mode, whose properties are all significantly affected by elasticity. For the first, the critical Marangoni number increases with elasticity. For the second, the work done by perturbation stress fluctuates in vertical direction. The last becomes the preferred mode when the elasticity is high enough and its perturbation energy comes from the Marangoni force caused by perturbation temperature while dissipates by perturbation stress. Their mechanisms are discussed and the comparisons are made with Newtonian fluid. (C) 2016 AIP Publishing LLC.