It seems that if (1.) the economy doesn't grow, and (2.) old people borrow from young people so they can simply consume more, it is trivially true that debt can be burdensome to future generations. This is true for any kind of debt - public or private. And it tells us very little about the way that economies in human societies actually work (which is our goal here as economic scientists, after all - we're here to understand human social behavior, not play around with OLG models).
I don't know why Nick is embracing a conclusion that is entirely reliant on unrealistic assumptions. When we're discussing something like a general glut Nick is always the first to point out how wrong this sort of approach is. So why is he doing it here? I don't know.
Nick often tells us that in the real world we trade with money and excess demand for money can spill over into other goods, causing a general glut. There is no market for money - money is traded in every market. Any model that abstracts from this is not reliable for dealing with the real world.
Well when we think about debt, we need to think about it in a world where the economy is growing and people borrow either as a result of time preference (they'll pay a premium to consume sooner) or expectations of a rate of return exceeding the interest rate. Under both of those more realistic assumptions, Nick's concerns about the necessesity of a burdensome debt break down (and Nick even admits this!) so why is he so emphatically saying that debt is burdensome and only non-burdensome under certain atypical conditions (rather than non-burdensome and only burdensome under certain conditions)? I just don't know.
Krugman is presumably not talking about a bizarre zero growth, all consumption, OLG world (that would be as silly as thinking about a barter economy when you theorize recessions). I always got the impression he was talking about the real world. In the real world income grows, and government invests in education, health, roads, institutions, research, infrastructure, defense, etc.. And in the real world, increasing debt levels means that one person with a growing income will, in the future, pay some money to another person with a growing income. There is no inherent reason why that debt relationship is a burden on the future. Distributional and incentive problems may apply, and of course problems could crop up if you grow the debt faster than you grow your ability to pay it. But the point remains that the unit of analysis (the community) encompasses both the asset holder and the liability holder, so nothing about the debt relationship itself is necessarily burdensome.
More interesting problems of course arise when you are in a situation that fits Nick's assumptions more closely: spending on the consumption of older members of society. If we want a simple transfer to older members of society because we think the elderly are deserving of such a transfer, then we'd want to prevent that program from being financed by deficits and we'd hire an actuary to make sure everything is above board. We've done exactly that with Social Security. If the costs of this consumption start increasing uncontrollably, then we're going to have to rethink that transfer payment as a society. We're doing exactly that (and should do more of that) with Medicare. But these are both questions of the social desirability of a certain type of transfer payment in a democracy. They have nothing at all to do with debt burdens because we don't pay for Social Security or Medicare by issuing debt. Indeed, Social Security and Medicare are two major holders of government bonds.
UPDATE: Nick's comment section on this one is voluminous, and I enter briefly on page three and get into an issue that's also raised in this comment section. If you're worried about Dean Baker saying that it's "impossible", then I agree with Nick Rowe that it's not "impossible" for debt to be a burden on future generations. On the question of Dean Baker's word choice I am with Nick. But I didn't even realize that was the real question (I certainly don't think it's the important question). I thought the question was "is public debt like personal debt - if we contract public debt does that mean that the public's future is burdened in the same way that when we contract personal debt our future is burdened by repayment?". The answer seems to me to be "no - it's a very different sort of thing", and we need to push back against this sneaking in of microeconomic thinking into macroeconomics.