I have a on premise power BI server and my source db is a on premise MYSQL server . i would like to do refreshing to pull data periodically, for that should i really required a on premise gateway. If it is required what are things i have to take care of its installations.
No, in this case Power BI Gateway is not required. It is needed, when on-premise data source needs to be used by cloud services (outside of your internal network).
What is an on-premises data gateway?
The on-premises data gateway acts as a bridge to provide quick and secure data transfer between on-premises data (data that isn't in the cloud) and several Microsoft cloud services. These cloud services include Power BI, PowerApps, Power Automate, Azure Analysis Services, and Azure Logic Apps. By using a gateway, organizations can keep databases and other data sources on their on-premises networks, yet securely use that on-premises data in cloud services.