How do tax payers have to pay for the tax deductions someone else receives due to a charitable donation?
Well you actually answered this question yourself. It is because of this:

While I do understand, that the government loses money in tax incomes, I don't understand how this directly increases the tax burden on other tax payers (except for, in the long term, the government might raise taxes to make up for lost tax revenues).

To be more precise an alternative is also for government to spend less but I think that the articles are arguing that ceteris paribus keeping government spending fixed government will have to somehow eventually collect the resources from economy in one way or other. In a long run government has to respect its intertemporal budget constraint. If someone pays less, taking government spending as fixed, someone will have to pay for that shortfall either directly or indirectly eventually at some point.