No. I work exclusively with scaling Linux workloads in Azure and local failover happens fairly regularly without the user ever seeing any indication. The hypervisor has its own cache which is tied more to your storage acct & runtime data-disks than the VM itself. This is true even with the lowest cost storage option: LRS standard blob-storage.

Howwwwever, LRS storage does not save you if the whole datacenter goes down, or during scheduled maintenance (when the whole datacenter is down.) For that, you'll need ZRS (which does failover to a co-lo in the event of the primary datacenter going down) or GRS (for which you can configure/test your failover options.)

Also, Microsoft's strength is in their PaaS services, like app-service or Azure-Functions. Those usually have CosmosDB on the backend, which is pretty much the best failover/DB-availability server-software on the market in my opinion.

VMs in all clouds are always tied to specific machines. If that machine fails unexpectedly then those VMs will restart. If it is a controlled reboot (e.g. host update) then they may not restart...