Let $x=b-a$ and $y=c-b$ so that $b=a+x$ and $c=a+x+y$. Then $$2^{a} 3^{b} 5^{c}=2^{a} 3^{a+x} 5^{a+x+y}=30^{a} 15^{x} 5^{y}$$ and $a, x, y$ are any positive integers. Thus $$\begin{aligned} \sum_{1 \leq a \leq b<c} \frac{1}{2^{a} 3^{b} 5^{c}} & =\sum_{1 \leq a, x, y} \frac{1}{30^{a} 15^{x} 5^{y}} \\ & =\sum_{1 \leq a} \frac{1}{30^{a}} \sum_{1 \leq x} \frac{1}{15^{x}} \sum_{1 \leq y} \frac{1}{5^{y}} \\ & =\frac{1}{29} \cdot \frac{1}{14} \cdot \frac{1}{4} \\ & =\frac{1}{1624} \end{aligned}$$