Several cloud-based applications, such as cloud gaming, rent servers to execute jobs which arrive in an online fashion. Each job has a resource demand and must be dispatched to a cloud server which has enough resources to execute the job, which departs after its completion. Under the `pay-as-you-go' billing model, the server rental cost is proportional to the total time that servers are actively running jobs. The problem of efficiently allocating a sequence of online jobs to servers without exceeding the resource capacity of any server while minimizing total server usage time can be modelled as a variant of the dynamic bin packing problem (DBP), called MinUsageTime DBP. In this work, we initiate the study of the problem with multi-dimensional resource demands (e.g. CPU/GPU usage, memory requirement, bandwidth usage, etc.), called MinUsageTime Dynamic Vector Bin Packing (DVBP). We study the competitive ratio (CR) of Any Fit packing algorithms for this problem. We show almost-tight bounds on the CR of three specific Any Fit packing algorithms, namely First Fit, Next Fit, and Move To Front. We prove that the CR of Move To Front is at most $(2μ+1)d +1$, where $μ$ is the ratio of the max/min item durations. For $d=1$, this significantly improves the previously known upper bound of $6μ+7$ (Kamali & Lopez-Ortiz, 2015). We then prove the CR of First Fit and Next Fit are bounded by $(μ+2)d+1$ and $2μd+1$, respectively. Next, we prove a lower bound of $(μ+1)d$ on the CR of any Any Fit packing algorithm, an improved lower bound of $2μd$ for Next Fit, and a lower bound of $2μ$ for Move To Front in the 1-D case. All our bounds improve or match the best-known bounds for the 1-D case. Finally, we experimentally study the average-case performance of these algorithms on randomly generated synthetic data, and observe that Move To Front outperforms other Any Fit packing algorithms.