In most cases, no. Hosts have different policies on upgrading plans, but a good host will understand that that the unexpected can happen and give their customers some leeway. Most are also more than happy to pick-up the larger tab.
Before signing up with a new host, ask about the upgrade policy, especially if renting a dedicating server. Make sure that the host has a number of faster servers and doesn’t charge customers to migrate. No one wants to pay a setup fee twice.