Note: Updated with more information!

It's a difficult question to answer for a variety of reasons. There isn't always data available from WWI and WWII about mental trauma. The data that does exist doesn't have much granularity - it doesn't single out flamethrowermen, platoon sergeants, or other men by their occupation. There was (and still is) a real reluctance by many veterans to talk about their experiences and the related traumas.

The best answer I can give you is this: Possibly, but probably not. WWI and WWII were extreme traumatic wars to fight, regardless of what weapon you carried.

To begin with, there are two common, but very different, types of battle-related psychological trauma. The first is combat stress reaction (CSR or COSR), the short-term, immediate response to traumatic battle experiences. Today, it's generally regarded as a normal (even helpful) response to the stress of combat, but it can become debilitating if it becomes too severe. Symptoms like hypervigilance, hyperstartling, bad dreams, and irritability are common in soldiers with undergoing CSR. With treatment and/or time, moderate cases of CSR will diminish and eventually go away. More serious cases of CSR can lead to serious problems down the road.

Most WWII cases of "battle fatigue" are likely cases of severe combat stress reaction. Some soldiers developed "thousand yard stare," one the best known symptoms of acute CSR. Soldiers slo simply "gave up" or "shut down." Some went even further - one G.I. recalled a squadmate in north Africa who “went crazy and beat his head against our foxhole till his skin on his forehead was just hanging in strands. He was foaming at the mouth just like a madman.”

While popular imaginings of battlefield trauma tend to focus on soldiers seeing horrible things like buddies blown apart or people burned alive, that's just one contributing factor to CSR. The 1994 edition of FM 22-51 states:

The severity of [CSR] symptoms and the speed and extent to which they respond to treatment are directly related to the intensity, lethality, and duration of the battle incidents which caused them.

These findings are based on research that goes back to WWII. Researcher John W. Appel went to Italy from May to July, 1944 and spoke with officers about how their men were handling the strain of combat. The officers had this to say:

"Most men ... were ineffective after one hundred and eighty or even one hundred and forty days. The general consensus was that a man reached his peak of effectiveness in the first ninety days of combat, that after this his efficiency began to-fall off, and that he became steadily less valuable thereafter until finally he was useless."

As several psychiatrists wrote in the 1946 "Combat Exhaustion" report:

“There is no such thing as ‘getting used to combat ... Each moment of combat imposes a strain so great that men will break down in direct relation to the intensity and duration of their exposure. ... Psychiatric casualties are as inevitable as gunshot and shrapnel wounds. ... The general consensus was that a man reached his peak of effectiveness in the first 90 days of combat, and that after that his efficiency began to fall off, and that he became steadily less valuable thereafter, until he was completely useless.”

In especially intense combat, men could break even faster. In the grinding, relentless hedgerow battles after D-Day, 98 percent of American infantrymen still alive after 60 days of fighting became battle fatigue casualties. These men were psychologically-debilitated, they "shut down" from combat stress and had to be pulled off the line. During the 82-day battle of Okinawa, there were 26,000 battle fatigue casualties (nearly one in ten combat troops). Indeed, the Pacific was especially stressful for soldiers and Marines, with higher rates of battle fatigue there than in the ETO or MTO.

Combat stress affected enormous numbers of soldiers in combat. In the ETO, 65 percent of surveyed American combat veterans admitted they'd frozen or been incapacitated by battle fatigue-related extreme fear. Nearly of all 40 percent of all medical discharges during WWII were psychiatric “Section 8" discharges. All in all, nearly one in four American casualties weren't physically wounded - they were battle fatigue victims.

So what does this mean for flamethrowermen?

Their combat experience was just as intense and as frightening as that of other grunts. When it comes to the intensity, flamethrower operators were in the same firefights as other infantrymen.

Combat may have been more lethal for some flamethrower men, especially in the Pacific, since they drew fire and had to close in with a dug-in enemy. Some anecdotal evidence supports this, but it has to be contextualized with the already high casualty rates of infantry units in WWII. In the 36 days of fighting on Iwo Jima, every single Marine infantry battalion took over 50% casualties. Just two months after D-Day, U.S. Army infantry companies in Normandy had taken 100 percent casualties. Nearly 90 percent of a U.S. Army infantry division's casualties were infantrymen and almost 70 percent of all American casualties in WWII were infantrymen, even though they made up only 6 percent of the U.S. Army (by far the largest of the combat branches).

As for duration, flamethrower operators, particularly those in the PTO, often fought on the line for about the same time as every other infantryman. That could mean month after month of intense combat - In Europe and Italy, most armies kept combat troops on the line for extended periods. Troops might be pulled of the line for a short R&R period, then thrown back into combat. The only ticket home was serious injury, hopefully a "million dollar wound" sent a man home, but didn't maim him.

Indeed, some flamethrower operators may have actually done less fighting than other combat units. This seems to have been more true in the ETO. The British, for example, carefully-husbanded their limited number of Crocodile tanks. Some flamethrower-using combat engineering units had responsibilities not related to direct combat (ex. repairing bridges or clearing mines), so they may have sometimes seen marginally less direct combat than a line rifle company.

A few flamethrower-using units even gave them up entirely, because they weren't using them much. Some U.S. Army units in the ETO got rid of their man-portable flamethrowers by late 1944 and early 1945. They'd found weapons bazookas were safer and more broadly useful than flamethrower.

So what's the takeaway from all this? The key CSR risk factors of combat duration, lethality, and intensity don't seem to have been dramatically or uniquely higher for flamethrower men.

The deciding factor for combat stress and battle fatigue wasn't occupation. It was rest. The "breaking point" for American riflemen in the ETO and MTO could be as much as 240 days, or as little as 140 days. British riflemen, on the other hand, could stay in combat for up to 400 days. Was Tommy tougher than Joe? Maybe. The critical difference seems to have been rest policies. When possible, British soldiers stayed on the line for about two weeks, then got around four days of rest. Meanwhile, the U.S. Army kept soldiers on the line for up to 80 days at a time. As a result, American troops burned out faster.