No, installing an eSIM will not lock your phone.
An eSIM (embedded SIM) is a digital SIM that is embedded in your phone's hardware and can be activated through the device settings. It does not have any physical component that needs to be installed or removed from your phone unlike a physical SIM card.
When you install an eSIM it will not affect the lock status of your phone. The lock status of your phone is determined by your primary SIM carrier (usually the provider of your device) and depends on whether or not your phone is locked to a specific carrier. Installing an eSIM does not change this status.