If the Na+ voltage gated channels remain open instead of getting deactivated during the re-polarization period, would the membrane potential become 0 since the Na+ ions would be constantly bringing into the cell while K+ is transported out of the cell, which makes them cancel each other out in a sense? However, I feel like there is something wrong with my reasoning but I don't know which part.
" If the Na+ voltage gated channels remain open instead of getting deactivated during the re-polarization period, would the membrane potential become 0 ? "
No, if Na+ channels remain open then the repolarization period would never happen.
approximate relative permeability values at the peak of a typical neuronal action potential are pK : pNa : pCl = 1 : 12 : 0.45 calculator
and thus we can see that the membrane potencial is much higher than the opposite for example
For a typical neuron at rest, pK : pNa : pCl = 1 : 0.05 : 0.45 calculator
and the only variable that we have changed was the relative membrane permeability for Na+
Hypothetically speaking if [K+]i = [K+]o = [Na+]i = [Na+]o = [Cl-]i = [Cl-]o the membrane potential in that case in fact would be zero.
However that is impossible because at rest, Na⁺/K⁺ ATPase constantly moves 3 Na+ ions out and moves 2 K+ ions in, and thus these gradients are maintain, so they cannot ever be the same.
Regarding your other question
What would happen to the membrane potential if a cell didn't have developed relative refractory period?
In that case the probability of repolarization and hiperpolarization to occur is diminished and thus it's possible to assume an extreme scenario of convulsions with death associated.
Example: Strychnine is a neurotoxin which acts as an antagonist of glycine and acetylcholine receptors
on the other hand, an excessive relative refractory period scenario, would result in a difficulty to achieve the threshold and depolarization
Example: mechanism of action of local anesthetics
by applying the principle above it is possible to reversibly decrease the rate of depolarization and repolarization of excitable membranes
As a side note (or curiosity), based upon that equation we can also modify Cl- values, and in that case we can speak about the mechanism of action of hypnotics and sedatives (such as benzodiazepines)