Since the compressed spring has stored energy the answer would be "Yes, but so little as to be completely unmeasurable." E=mc^2 applies to all energy and mass,

Since the compressed spring has stored energy the answer would be "Yes, but so little as to be completely unmeasurable." E=mc^2 applies to all energy and mass, but if you work the equation with sample energies, you will quickly discover that a substantial amount of energy becomes very, very little mass.