Nonlinear energy sinks (NES) with broad energy absorption capabilities and inerters with significant mass amplification effects have been extensively studied for vortex-induced vibration (VIV) mitigation. However, these studies often overlook the sensitivity of vertically nonlinear vibration systems to additional load effects, particularly in relation to weight changes. To address this gap, the present study delves into the impact of weight effects on the VIV mitigation performance of nonlinear energy sink inerter (NESI). A generalized model of the NESI-beam system, accounting for weight effects, was developed and numerically solved. The findings reveal that the VIV control effectiveness of an ideal NESI (without weight) deteriorates under weight effects due to significant changes in the effective control parameter distribution. Employing a low inertance conversion ratio can significantly mitigate this limitation. Additionally, the enhanced NESI system balances the weight through a positive linear stiffness element, thereby avoiding the influence of weight on the nonlinear stiffness system and achieving complete elimination of weight effects.