Basic flame weapons are effective against flammable targets or in very close quarters. Normally they are not used in any situation where the defenders would have a clear view of the attackers. Flame weapons are currently popular among insurgent fighters due to their simple operation and effectiveness in ambush situations.
In World War II, flame throwers were used to clear out no-surrender fortified positions which were primarily found in the Pacific theater of war. Normally when a line or position is overrun the defenders will retreat or surrender. This was not the case for Japanese forces which would often hole up in pillboxes or bunkers and fight to the end. Against such positions a flame thrower is effective if the attacker can approach it from the side or the rear and incinerate it. When the defensive line is intact this is not usually possible, so flame throwers were primarily used in "mop up" operations to destroy fortifications that had already been strategically nullified.
In modern warfare, flame throwers are considered to be largely superseded by thermobaric weapons or incendiary rockets which are more powerful and work at longer ranges. Thermobaric weapons are useful in urban or other close quarters in which it is either (1) difficult to target projectile weapons, or (2) the situation is so close, that the use of projectile weapons could harm friendly forces accidentally.
The US equivalent is the Flash rocket launcher. American forces rarely use the Flash due to the high probability of civilian casualties if the weapon is used in an urban environment. Modern military soldiers also have access to incendiary grenades which can be used to ignite flammable materials. These weapons can be effective against pillboxes or other positions which have a limited field of view. Normally such positions are no longer tactically relevant because modern artillery and rockets are so accurate and powerful that pillboxes can be destroyed from a long range.