In this study, we characterize the shear and extensional rheology of dilute to semidilute solutions of cellulose in the ionic liquid 1-ethyl-3-methylimidazolium acetate (EMIAc). In steady shear flow, the semidilute solutions exhibit shear thinning, and the high-frequency complex modulus measured in small amplitude oscillatory shear flow exhibits the characteristic scaling expected for solutions of semiflexible chains. Flow curves of the steady shear viscosity plotted against shear rate closely follow the frequency dependence of the complex viscosity acquired using oscillatory shear, thus satisfying the empirical Cox-Merz rule. We use capillary thinning rheometry (CaBER) to characterize the relaxation times and apparent extensional viscosities of the semidilute cellulose solutions in a uniaxial extensional flow that mimics the dynamics encountered in the spin-line during fiber spinning processes. The apparent extensional viscosity and characteristic relaxation times of the semidilute cellulose/EMIAc solutions increase dramatically as the solutions enter the entangled concentration regime at which fiber spinning becomes viable.