I am a physicist and GPS indeed takes relativistic effects into account. GPS would not just be slightly wrong if it didn't: it would be entirely worthless. See for example [1].
Fun fact: When designing the original system there was disbelief amongst some involved that the relativistic effects were either real or relevant so they built in a switch to be able to remotely disable those parts of the calculations. They never did disable them.
[1] http://www.astronomy.ohio-state.edu/~pogge/Ast162/Unit5/gps....