It is common wisdom that altruism is a crucial element in addressing climate change and other public good issues. If individuals care about the welfare of others (including future generations) they can be expected to unilaterally adapt their behaviour to preserve the common good thus enhancing the wellbeing of all. We introduce a network game model featuring both altruism and a public good (e.g. climate) whose degradation affects all players. As expected, in an idealistic fully connected society where all players care about each other, increasing altruism results in a better protection of the public good. However, in more realistic networks where people are not all related to each other, we highlight an intrinsic trade-off between the effects of altruism on reducing inequality and the preservation of a global public good: the consumption redistribution generated by a higher altruism is partly achieved by lowering income transfers towards protection of the public good. Therefore, it increases overall consumption and is thereby detrimental to the public good. These results suggest that altruism, although good from a welfarist point of view, is not in itself sufficient to simultaneously solve public good and inequality issues.