It might affect. If you have issues with the ranking after moving, checkthe search engine status of your domain. Usually **it happens when domain is transferred.
No they should not unless you have huge downtime. If you are about the moving to the new server I would recommend you think about good web host to go with. I recommend you have a look at just199.com
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Yes it will be affected, mostly your PR might become none for a moment until google finish reevaluate your website then it will back to it's original state.