Is this grapefruit tree planted too close to my yukka?

It's very close so I'm wondering if this will affect the grapefruit's growth. I'm ok to cut the yukka down and cover the base to try kill the roots, but only if it's necessary.

Short answer is yes. The yucca has a shallow root system that is designed to suck up all the water that lands in its root zone so any tree within 30-50 feet of it will die of thirst.
As an average rule of thumb no tree should be planted closer to any other tree than 15 feet, this allows for adequate root zones for all trees except those desert trees like the yucca and cacti.

Redhawk

Agree with Redhawk.  Both trees have similar rooting characteristics—shallow, fine roots for most citrus trees.  That poor grapefruit will never thrive in that location.  Yucca are tough, strong plants.  They thrive in places where other plants struggle.  Grapefruit (and most citrus as well) are also tough, but why put them in a place where they'll have to engage in a fight to the death?  You want them cooperating, not competing.

The other thing I've found with various species of yucca plants is that their roots can be invasive, and can even be sidewalk busters.  I'd never plant one close to any sort of hardscaping or foundation.  I've seen yucca lift a block wall with an 18 inch footing and lean it over in less than 7 years.  Once they get their roots under something, it's going to come up.  I see people planting them next to their house and I think, "You're basically planting a hydraulic jack next to your foundation.  This will not turn out well."

That said, there are dozens of different kinds of plants that commonly get called Yucca.  I'm not familiar with that specific plant (and there isn't enough of it in the picture to clearly identify it) but my hunch is that it's not compatible with anything being planted close to it.