If he is actually drinking from the mug and then topping it up and warming it....then, yes, bacteria will flourish - but not necessarily "harmful". (The same applies when people drink bottled water from the bottle, re-cap it and leave it in a warm place eg car...the bacterial count shoots up).
My main reservation with doing this....the water becomes less palatable, because it loses dissolved gases (like oxygen and carbon dioxide) and tastes "lifeless" or flat.