The tetragonal fluoro-scheelite LiREF4 compounds (RE=rareearth) have been shown to exhibit a variety of useful optical and magnetic properties. While LiYF4 has been widely studied, many of the fundamental thermodynamic properties of other members of this family remain unknown. Here, we report the complete elastic tensors (Cij) of single-crystalline LiYF4, LiTbF4, and LiGdF4 using resonant ultrasound spectroscopy and density functional theory (DFT). We compare the results for LiYF4 with prior experimental results using time-of-flight ultrasound methods. This is the first report, however, of the experimental elastic tensors of LiTbF4 and LiGdF4. The present results point to a softening of the elastic moduli of the LiREF4 system when Y is replaced by the larger ionic radius of Tb or Gd. Furthermore, we find that just 0.3% doping with Nd on the Y site also leads to a slight softening of the moduli. The variation of the elastic moduli as a function of temperature up to 216∘C was also measured. A nearly linear softening of all seven independent elastic moduli was observed with increasing temperature. Phonon dispersions and phonon density of states obtained by DFT support the experimental finding of a significantly higher sound velocity due to lighter Y atoms in LiYF4, as compared to heavier LiTbF4 and LiGdF4.