Which would be safer: removing hydrogen or adding hydrogen to our sun?
My kneejerk reaction is that your only option is to remove a chunk of mass from the outer part of the Sun.

The Sun will respond (on a Kelvin-Helmholtz timescale), by contracting and becoming less luminous because the core temperature is lower in a less massive star. This will extend its main sequence lifetime, because only the central parts of the star are involved in fuelling the nuclear burning. The core is separated from the well-mixed outer convective envelope by a stable radiative zone.

Since the default solar evolution involves it getting slowly more luminous with time, it is possible that mass could be extracted at just the right rate to keep the Sun at constant luminosity. This sounds like an interesting calculation to do.

The alternative of adding "fuel" in the form of hydrogen won't work. The Sun's luminosity will increase because of its increased mass and increased central temperature. However, because the new fuel cannot be mixed into the core (only the outer part is mixed by convection), then the main sequence lifetime would be reduced.