No, Human Green Lanterns do not become green when they are utilizing their powers. However, they can exhibit a green force field, particularly when they are in space or in other hostile conditions which could account for a green tint.
In the image below, the thick bright outline around Hal Jordan is his forcefield. However, the skin of his face is a regular human skin colour, except for his trademark green mask.
- Different costumes for Green Lanterns can also give the impression (particularly because of their skin-tight nature) that their skin has turned green when they activate their costumes.
These gentlemen are/were Green Lanterns of Sector 2814 of which Earth is a member. Left to right: Guy Gardner, Hal Jordan, John Stewart and Kyle Rayner. All four are wearing black and green costumes, with very little actual skin shown. Guy's wearing a white belt. Everyone's wearing white gloves except John Stewart. Hal Jordan and Kyle Rayner are wearing green masks.
- As for the Green Lantern movie: The costume design there had a very skin-like appearance even replicating muscle groups in its overall textured design. If you were potentially colorblind, it COULD resemble musculature and in effect skin.