Graph invariants play a crucial role in understanding the structural and combinatorial characteristics of graphs. The fault-tolerant metric dimension, as a significant graph invariant, finds applications in diverse areas such as robust network optimization, autonomous robot navigation and intelligent sensor systems. In this paper, we investigate the fault-tolerant metric dimension and fault-tolerant edge metric dimension of zero-divisor graphs arising from upper triangular matrices over a finite commutative ring. The obtained results contribute to the understanding of metric-based fault tolerance in algebraically structured graphs.