Yes, it seems that the two boys die in the gas chamber. Once Bruno has his hair cut off and he dons the "striped pajamas," there is little apparent difference among him and the Jewish prisoners who are marched into large, dark room that is airtight.
In Chapter 16 Bruno has lice in his hair and must have his head shaved; as a result, he more closely resembles Shumel. Then, in Chapter 17 Bruno learns that everyone in his family but his father are returning to Berlin, so he goes to see Shumel in order to inform his friend that he will be leaving. Outside the fence that confines Shumel, Bruno is saddened that they have not had an adventure. Then, he has a "brainstorm" and tells Shumel that if he could come inside the fence, he could help him look for his father as the boy is very worried about what has become of his parent. Shumel says that he knows where the uniforms are kept and that he will try to take one.
The next day, then, Shumel provides Bruno with a uniform, and after donning it, Bruno crawls under the fence. Bruno's next words, a quote of his actress grandmother's, ring with a terrible irony:
"You wear the right outfit and you feel like the person you're pretending to be."
Together, then, they search to no avail for Bruno's father. Just as Bruno is about to return to the spot where earlier he crawled under the fence so he can now return home, a loud whistle blows, and ten soldiers surround them. Because he has the outfit that makes him look like a Jew, Bruno is herded with Shmuel into a long room. Suddenly, everyone gasps as a loud metallic sound is heard from the outside.
From this description, the reader infers that the boys and others are sent into a gas chamber. For days, Bruno's father searches for him, but all he finds are his clothes by the fence; since there is no sign of Bruno anywhere on the camp or in the villages or towns nearby, Bruno's father deduces the terrible fate that befell his son.