It is well known by the virtual hand illusion (VHI) that simultaneous and synchronous visuotactile sensory feedback within a virtual environment elicits the feeling of ownership of a virtual hand, by observing for some seconds in a scene a virtual hand being touched while at the same time receiving tactile stimulation on the real hand in the corresponding positions. In this paper, we investigate possible modulations in the feeling of ownership (sensation of owning a virtual hand) and of agency (sensation of owning virtual movements and actions) according to whether or not the participant's own motor acts (1) induce coherent self-activated visuotactile sensory stimulations; and (2) generate plausible consequences in the simulated environment. For this purpose, we elicited the VHI within a group of participants through a cross-modal integration of visuo-tactile sensory stimulations within a dynamic and physically plausible immersive virtual environment, where they were able to perform natural tasks in both passive and active agency conditions. Our results indicate that both feelings of ownership and agency can be achieved in immersive virtual environments, when the subject is realistically interacting and performing natural upper limb movements. We did not observe any significant difference in the VHI in terms of ownership and agency between the active and passive conditions, but we observed that a physically incongruent simulated interaction with the virtual world can lead to a significant disruption of ownership. Moreover, in the passive agency condition, a plausible physical behavior of the virtual hand was sufficient to elicit a partially complete sense of ownership, if measured in terms of proprioceptive drift, even in the presence of an asynchronous visuotactile sensory feedback. All these findings suggest that the multisensory feedback associated with a subject's own actions and the physical plausibility of the environment both act as determinant factors, influencing and modulating the vividness of the VHI.