If the new hosting plan and server configuration is not optimal for your specific requirements then it is possible that you may experience some negative repercussions of switching.
Your new hosting provider and/or web developer / IT dept. should be able to assist in making sure the switch over to new hosting / server(s) is as smooth as possible.
It is best to leave your hosting plan intact and up and running for a good month after you have switched hosting just in case any DNS is resolving to the IP address of the server instead of / as well as the domain name(s). This way, you have a full month of DNS resolving to old server and new server before you terminate the old.