That depends if you mean PTSD in a clinical sense, or PTSD in the pop-culture-phrase sense. And also, you're asking a yes/no question, but it's probably better to ask "how likely is it". There are a lot of humans in existence, so a whole lot of really unlikely things are technically possible just by the laws of big numbers.
In the pop culture sense, PTSD is basically just residual stress/anxiety from some negative event that happened in the past. Burnout, or even generalized anxiety disorder with work-related triggers... these are all totally legitimate things that have a very negative toll on quality of life, and that people conversationally might refer to as PTSD, but from a clinical standpoint, they aren't the same.
PTSD has very particular clinical diagnostic criteria [1]. If you read carefully, they're perfectly understandable. But I want to make some key points: in a clinical sense, the trauma referred to in "post-traumatic stress disorder" is explicitly and exclusively "actual or threatened death, serious injury, or sexual violence". That's it, end of list. Among other things, a PTSD diagnosis requires some kind of repeating, severe negative episode(s) -- night terrors, dissociative reactions, panic attacks, that kind of thing. It has to last more than a month, and it has to cause significant disturbance in your life. PTSD is a disorder, but it is almost always a qualifier for a disability, because it is extremely common that PTSD precludes your ability to live a normal life. PTSD, in the clinical sense, is something I would find extremely, exceedingly rare in any kind of business situation, startup or otherwise.
To be clear, I don't want to be relativizing either PTSD or burnout or anxiety disorders. I don't want to be comparing anything; I've yet to see anything productive come out competitions about who has the worse health problems, mental or otherwise. There are plenty of other very real conditions that can and very likely do result from the stressful environment of a startup, and they can be very serious, even leading to suicide, and I really don't want to trivialize that. But I want to make one thing clear: from a clinical standpoint, PTSD is a distinct entity with concrete characteristics. I've personally been dealing with pretty severe mental health issues my entire life, and I've also had friends with diagnosed PTSD, and to be completely honest, even though I doubt you meant anything by it, conflating the two feels deeply disrespectful to all of us.
[1] https://www.ncbi.nlm.nih.gov/books/NBK207191/box/part1_ch3.b...