Even the real thing would be somewhat ineffectual on a zombie. They don't notice they are on fire, and human bodies don't burn well, being 70% water. While a flamethrower might be good against normal undead who are dried out and only magically ambulatory, zombies more or less operate the way people do. Things that would disable a conscious person DON'T stop zombies. And lots of contemporary military hardware isn't really about smashing the brains of people - it is about messing up soldiers and making it hard to do their jobs. Take land mines, for instance - they seem dangerous because we always see pictures of these sad-eyed children with missing limbs, but THEY ARE STILL ALIVE. And zombies are even harder to kill than children! If landmines can't reliably kill children who have military training, how do you expect them to stop the walking dead? Some mines are even meant to target your cojones which is the very least effective way to harm a zombie.

Flamethrowers are no better. They were used primarily to get at soldiers who were fortified in difficult-to-access positions. A zombie in such a position is either effectively incarcerated, or will leave to bite you, and thus render the flamethrower unnecessary. Flamethrowers are seen as horrible because they hurt a lot, and make ammunition blow up. Zombies are unarmed and don't care if they are on fire. Use one in a crowed area or tight quarters, and you have just made the zombie that much more dangerous, because now it can stagger about setting fires or burning the people it grabs even if they DO manage to evade the bites.

Also, the primary point of the original poster was that military rifles are difficult to obtain. How do you propose he go about obtaining a flamethrower?