Here is an attempt to clear up some of the confusion that still exists on this issue.
1) Government debt can be used to redistribute income to current generations from future generations, even if the aggregate level of consumption in each period remains the same. Proof by example: see here (or many similar proofs from Nick Rowe - his latest post on this is here).
Note: I think people get confused because although the first part of the proposition seems intuitive (if the government cuts taxes and pays for this by borrowing, surely those receiving the tax cut could spend it on themselves and be better off as a result), the ‘even if’ part seems wrong (if we are taking from the future to give to the present, current consumption must rise and future consumption fall). Those who have worked with OLG models, like Roger Farmer, find it easier to work through the logic as to how this is possible.
2) The size of government debt is not a good indicator of any burden. It is possible that government debt is positive, but there has been no attempt at intergenerational transfer. Proof 1: taxes on the young are cut, and the young save all the tax cut by holding the extra debt. Proof 2: borrowing for a capital project that benefits current and all future generations equally.
Note: This is important, as Noah Smith notes in this post. The size of government debt is not equal to the ‘burden’ on future generations. Indeed, positive debt is compatible with there being no burden at all.
3) There are probably much more important mechanisms going on right now that are transferring consumption from the future to the present: in some countries rising house prices, and climate change. No proof, just an opinion.
Note: despite this, if you think your grandchildren will have such a wonderful life compared to yours because of technological progress, you might not be too bothered. It is also worth remarking how potentially inconsistent it is to argue that we have to reduce debt now for the sake of future generations, and at the same time argue that it is too costly to take action now to mitigate climate change.
4) Even if no intergenerational transfer is involved, high government debt could reduce future consumption for two quite plausible reasons: productive capital may be crowded out, and the tax required to pay the interest on the debt is distortionary (i.e. reduces output below optimal level). Proof: countless papers in the literature.
Note: I think it is wrong to describe both mechanisms as model specific, because you have to make quite extreme assumptions to avoid them. It is for this reason that I worry about high government debt in the long term. I have not heard anything to convince me that either mechanism is unimportant, or of any countervailing mechanism. (The need for safe assets could argue for high gross government debt, but not net debt, where the difference could be a large sovereign wealth fund.)
5) None of these arguments justify austerity at the Zero Lower Bound. Proof: countless posts by various people, including myself.
Note: For example, crowding out happens through high real interest rates, which are hardly a current problem. Nor is scarcity of labour arising from tax distortions.
Final thought. Think about government debt as a way of providing intergenerational insurance against negative demand shocks. When those shocks happen, the state pays out by cutting taxes (say) and increasing its debt. In normal times taxes rise again and the debt is gradually reduced. In this case allowing debt to rise following one of these shocks is no burden.