Short Answer
This would create a new equilibrium, albeit different from potassium's Nernst potential, but still an equilibrium.
The Nernst equation is how you determine the voltage at which an equal number of potassium ions move in each direction across the membrane; that seems to be what you are calling "equilibrium" for potassium. If you have two ions and you'd like them both to have zero net flow, their Nernst potentials must be identical. The Nernst potential depends on relative concentration; the only way to change Nernst potential is to change relative concentration.
The Goldman equation will help you find a different "equilibrium": the voltage at which net total flow of all ions equals zero. This gives you the membrane potential with multiple ions present; however, at this equilibrium, as you've read, individual ions are not at equilibrium; for a typical neuron, you can expect that at the resting potential given by the Goldman equation you will have about equal potassium leaving the cell as sodium entering.
Longer Answer
In a typical neuron, there is more potassium and less sodium inside the cell relative to outside.
This makes a negative Nernst potential (=reversal potential) for potassium: to stop net flow of potassium down its concentration gradient and out of the cell, you need negative charge inside.
Similarly, there is a positive Nernst potential for sodium: to stop net flow of sodium down its concentration gradient and into the cell, you need positive charge inside.
The resting membrane potential is a weighted sum of these potentials that are driven by concentration gradients. The relative contribution of different ions to the membrane potential depends on their permeability; the more permeable the membrane is to an ion (through specialized channels), the more important that ion is for setting the overall membrane potential.
For a typical neuron, the resting membrane potential is near the potassium reversal potential, but not quite as negative, due to sodium (and other ions, but for now we can just think in terms of these two).
Let's say, for example, potassium reversal is -90 mV, sodium reversal is +50 mV, and the membrane potential is around -70 mV. The reason it's around -70 mV would be because that's precisely the voltage where as many potassium ions are flowing out of the cell as sodium ions are flowing in.
If pumps keep the concentrations of sodium and potassium constant inside and out, this will stay the same indefinitely: concentrations stay the same means reversal potentials stay the same which means the membrane potential stays the same, as long as ion permeabilities stay the same.
However, with no pump, the sodium concentration inside is rising, and the potassium concentration is falling. That means the reversal potentials won't stay the same. As the potassium concentration in and out becomes more similar, the Nernst potential for potassium drifts towards zero: you don't need any voltage to stop net potassium from flowing if the concentration is equal inside and out. As the sodium concentration in and out becomes more similar, the Nernst potential for sodium drifts towards zero. These potentials are not intrinsic to the ions, they are entirely a function of relative concentrations inside and out. If you lose the relative concentration gradients, you lose the voltage, and, importantly, you lose the ability to influence membrane voltage by changing the permeability of particular ions.
