Load balancing is among the basic primitives in distributed computing. In this paper, we consider this problem when executed locally on a network with nodes prone to failures. We show that there exist lightweight network topologies that are immune to message delivery failures incurred by (at most) a constant fraction of all nodes. More precisely, we design a novel deterministic fault-tolerant local load balancing (LLB) algorithm, which, similarly to their classical counterparts working in fault-free networks, has a relatively simple structure and guarantees exponentially fast convergence to the average value despite crash and omission failures. As the second part of our contribution, we show three applications of the newly developed fault-tolerant local load balancing protocol. We give a randomized consensus algorithm, working against $t < n / 3$ crash failures, that improves over the best-known consensus solution by Hajiaghayi et al. with respect to communication complexity, yet with an arguable simpler technique of combining a randomly and locally selected virtual communication graph with a deterministic fault-tolerant local load balancing on this graph. We also give a new solution for consensus for networks with omission failures. Our solution works against $t < \frac{n}{C\log{n} (\log\log n)^2}$ omissions, for some constant $C$, is nearly optimal in terms of time complexity, but most notably -- it has communication complexity $O((t^2 + n)\text{ polylog } {n})$, matching, within a polylogarithmic factor, the lower bound by Abraham et. al. with respect to both terms depending on $t$ and $n$. Ours is the first algorithm in the literature that is simultaneously nearly optimal, in terms of $n,t$, with respect to both complexity measures, against the adaptive omission-causing adversary.