We study lifelong reinforcement learning (RL) in a regret minimization setting of linear contextual Markov decision process (MDP), where the agent needs to learn a multi-task policy while solving a streaming sequence of tasks. We propose an algorithm, called UCB Lifelong Value Distillation (UCBlvd), that provably achieves sublinear regret for any sequence of tasks, which may be adaptively chosen based on the agent's past behaviors. Remarkably, our algorithm uses only sublinear number of planning calls, which means that the agent eventually learns a policy that is near optimal for multiple tasks (seen or unseen) without the need of deliberate planning. A key to this property is a new structural assumption that enables computation sharing across tasks during exploration. Specifically, for $K$ task episodes of horizon $H$, our algorithm has a regret bound $\tilde{\mathcal{O}}(\sqrt{(d^3+d^\prime d)H^4K})$ based on $\mathcal{O}(dH\log(K))$ number of planning calls, where $d$ and $d^\prime$ are the feature dimensions of the dynamics and rewards, respectively. This theoretical guarantee implies that our algorithm can enable a lifelong learning agent to accumulate experiences and learn to rapidly solve new tasks.