Thin liquid sheets are ubiquitous in many industrial processes, such as atomization and curtain coating. In the latter, the thickness of the deposited layer is limited by the breakup of the liquid curtain below a critical flow rate. The stability of a liquid sheet depends on the disturbance characteristics, sheet thickness and fluid properties. Experimental analyses have shown that thinner stable liquid curtains can be obtained with viscoelastic liquids; however, the underlining physical mechanisms associated with the increased stability are not fully understood. This work presents a numerical analysis of the effect of viscoelasticity on the stability of a thin liquid sheet. We first derive linear stability criteria for both planar and axisymmetric perturbations of Newtonian and Oldroyd-B liquids. The time evolution of planar and axisymmetric perturbations in an Oldroyd-B liquid sheet is evaluated using asymptotic expansion of the flow variables and a fully-implicit time integration scheme. The breakup time is calculated as a function of Deborah number and polymer to solvent viscosity ratio. The results show that the liquid rheological behavior does not change the linear stability criterion, however it has a strong effect on the growth rate of the disturbance and consequently on the breakup time.