It is looked for $\int_B xyz ~$ where $~ B=\{0<x<1, 0<y, 0<z, 0<y+z<1\}$.
Is it possible to write the limits this way?
$$\int_0^1\int_0^1\int_0^{1-y}xyz~dz~dy~dx$$
After some calculation I mistakely got -11/48 as the answer (1/48 is correct).
So is it correct to say that $z$ goes from $0$ to $1-y$ in the first place?
Your integration domain differs in your topic and your question text. $$\{0<x<1, 0<y, 0<z, 0<x+y<1\} \not= \{0<x<1, 0<y, 0<z, 0<y+z<1\}$$ For your second domain your ansatz is right and it holds $$\int_0^1\int_0^1\int_0^{1-y}xyz~dz~dy~dx = \frac{1}{48}$$ So you just miscalculated the integrals.
Show your calculations to get your mistake.