can a vampire Overwhelming Soul kineticist use Mental Prowess (Su) to gain negative levels to increase its power since it can't use burn??
Nope because his undead trait give him immunity to level damage/drain
Usually, abilities tell you if becoming immune negates the benefit.
In this case there is no such text and the negative happens only when forced to take burn against their will. Its not the first time that undead would gain an extra benefit. For example dealing with poisons while immune to said poison.
Yeah but the vampire can not suppress the immunity and if you can't gain negative lvl the ability can't apply
Mental Prowess (Su): An overwhelming soul's mind is strong enough to protect her body from the stress of channeling the elements. However, in exchange, she is unable to push her limits quite as far as other kineticists. She can't choose to accept burn, though she can use wild talents with a burn cost if she can reduce that cost to 0 points with abilities such as gather power and infusion specialization. If she would be forced to accept burn against her will (for example, if she fails a concentration check), instead of taking burn she takes 1 temporary negative level, which never kills her and can't be removed with restoration or greater restoration, but which disappears automatically after 24 hours. At 6th level, once per day, she can reduce the total burn cost of any wild talent by 1. She can use this ability an additional time per day at 10th level and every 4 levels thereafter. This ability alters burn and replaces internal buffer.
Since this ability is altering burn, it should conform to the rules of burn except where it is changed. One of the rules of burn is that a Kineticist that is incapable of taking non-lethal damage cannot accept burn. Mental Prowess alters burns so that instead of taking non-lethal damage you gain negative levels. If you are not capable of gaining negative levels, you cannot use Mental Prowess in this manner.
can a vampire Overwhelming Soul kineticist use Mental Prowess (Su) to gain negative levels to increase its power since it can't use burn??
Negative levels, in and of themselves, do not affect undead targets. Some methods of inflicting them DO have specific benefits for undead targets, such as the temporary HP "inflicted" by enervate and energy drain; but those are always called out.
When an Overwhelming Soul fails a concentrate check while gathering, the "cookoff" inflicts negative levels in lieu of burn. The OS cannot choose to accept burn, even as negative levels; they only result from this cookoff effect. These negative levels also don't have special properties if an undead suffers cookoff; they simply won't be inflicted.
Thus, the answer is no. Even if we suppose that a vampire OS can intentionally induce cookoff, there would be no effect at all. No drawback, no benefit.