In the case one does so while making dhikr in a group and people dislikes that act and are not accustomed to it, then you should not do so. But if people do not dislike this act, then there is nothing wrong with this act especially if the chewing gum is small. However, it would be preferable to avoid doing that while making dhikr in a group.

But while making dhikr alone, one is permitted to do so while standing, sitting, or sleeping on his side. And, there is nothing wrong to make dhikr while chewing gum as long as he does not consider this as a kind of disrespect towards the act of making dhikr. May Allah reward you and grant happiness in this worldly life and in the Hereafter.