What is the danger of (China's) investment-led growth as opposed to consumption-led growth? I keep reading that China must switch to consumption-led growth or productivity-led growth, that investment-led growth is 'dangerous.'

I understand that it is probably not sustainable in the sense that returns on capital will start to fall after awhile. But what else is the danger?
From the Chinese perspective, it's "dangerous" because high investment for so many years, particularly when driven by state incentives or by state-owned firms might induce mis-allocation of resources, high indebtness levels and excess capacity (some might say even voluntary, for example dumping in steel to hurt developed countries' industries). For instance, this paper states that:


  The main  objective  of  management  in  Chinese  state-owned  and  state-
  dominated  firms  has been  to  maximise  not  profits  but  the  growth  of  investment  and  output.  The  reward  to such  bureaucrats  takes  the  form  of  prestige,  power  and  the  accompanying  perks  of commanding an organisation; the reward being greater the larger the organisation.


Another reason is that investment is more volatile than consumption. However, if investment is supported importantly by state-owned enterprises, this might not apply well to China, which by the way have not had a recession in decades).

From a western perspective, high investment (and low consumption) contributes to a high trade balance surplus and therefore to a high current account surplus, which might be good for the central bank of China (accumulation of reserves) aiming to keep the Yuan relatively low to foster the competitiveness of the export sector, but is bad news for the rest of the world. The latter would benefit from higher consumption of the Chinese population (via Chinese imports and rest of the world's exports). The same story is told of Germany, which would help other countries (mainly European partners, because of same currency) by rebalancing its economy from exports toward more consumption. 

Additionally, note that some economists have blamed China (among others) for the Great Recession (e.g. here or here), partly because of their voracious reserve accumulation policies, a core element in the export-led Chinese development strategy. Thus, a more rebalanced global economy might be good for the world economy as a whole, decreasing the likelihood of another recession.