What density of an object of more than one substance affects the buoyant force?

Usually, by buoyant force, people mean the upward Archimede's force equal to the weight of the same volume of the body filled by the surrounding liquid. It is the global effect of the surface force due to a gradient of pressure in the surrounding liquid.

Therefore, the substance the body is made of and the container's weight do not enter in the buoyant force expression. They do enter the body's weight, which is the force that has to be compared with Archimede's force to establish the buoyancy of the body.

If we immerse a sealed cup of a liquid in a fluid, what density is taken in consideration when it comes to the buoyant force, the density of the cup itself (the substance in contact with the fluid) or the whole cup (with the liquid inside)?

Although it should be noted that in many buoyancy examples the mass of the container itself is neglected, technically you need to account for both the container and its contents in order to determine the volume of the fluid displaced, which, in turn, determines the buoyant force.

In order to determine the upward buoyant force, you need to determine the volume of the fluid displaced by the cup+liquid. To do that, you need to determine the density of the combination of the cup+liquid. To calculate the density, take the sum of the mass of the cup and the mass of the liquid and divide by the overall volume of the cup+liquid. That gives you the density of the combination of the cup+liquid.

If that density is equal to or greater than the density of the fluid in which it is immersed, it will either float completely submerged or sink. Either way, the upward buoyant force will be the same and will equal the weight of the fluid having volume equal to the volume of the cup+liquid.

On the other hand, if that density is less than the density of the fluid, the cup+liquid will float partially submerged. The submerged volume of the object will equal the volume of the fluid displaced, $V_f$, or

$$V_{f}=V_{cup+liq}frac{ρ_{cup+liq}}{ρ_{f}}$$

Where $rho _f$ is the density of the fluid. The upward buoyant force will then be

$$F_{buoyant}=rho _{f}V_{f}g$$

Hope this helps.

Neither one nor the other of the two densities are taken, but rather both. For example, say the bottle had a mass of 5 g and volume (plastic only) of 10cm³, the bottle would have a density of 0.5g/cm³. And say the liquid inside had a mass of 100g and volume of 100cm³, the liquid would have a density of 1g/cm³. However, when combined, they have a density of 0.95454... g/cm³ (105g/110cm³). However, as you can see, this is extremely close to the density of the liquid (1g/cm³), which is why the bottle density is often ignored, as it has a negligible effect upon the density.

Technically, however, it is this total average density that should be used to calculate buoyant force.