One example, in the US: poor people get crappy education. The more our economy shifts to a knowledge-based economy, the more effective this is at keeping poor people poor.

Another example: access to legal services (in the US) is essentially required for fair treatment under the law. Poor people cannot afford legal services, and so will generally not get fair treatment under the law. So a mistake/misunderstanding/relatively minor infraction that would be a mere hassle for a richer person can be life-ruining for a poorer person. In the US, we've gone so far as to apply wildly different penalties to "poor people's drugs" and "rich people's drugs" (obviously stereotypes, but the stereotypes have some basis in fact and are why the laws were written that way).

Poor neighborhoods are more likely to experience elevated crime levels. In the US, this means they are often militarized. This makes everyone that lives in them more at risk for life-ruining run-ins with the law. The people of NYC have decided that it's fair for non-white juveniles to be subject to stopping & frisking anytime, for no cause. This is a tax for which no benefit is derived by the payers.

There are tons of examples like these that have nothing to do with redistribution. If you read the papers, you'll see examples every day.