If I reincarnate and civil damages persist, does the expected future value become infinite? Or is there a way to deal with this.
Your hypothetical is a non-sequitur, whence it is not amenable to legal remedies.
Legislation does not contemplate the notion or possibility of [natural person's] reincarnation. Or at least you don't point to a specific legal system where the concept of reincarnation is admissible.
Even if a legal system acknowledged the concept of reincarnation, one ought to look at that system's doctrines to ascertain the meaning, purpose, and process of reincarnation. For instance, consider a legal system premised on the idea that the purpose reincarnation is a do-over that frees the individual from the burden of his experiences from previous lives. In that legal system, adjudging and enforcing any "persisting" damages would contravene that purpose because the process inevitably prompts that individual to wrap his head around matters that are supposed to be wiped from his memory and emotions.
In many situations, the notion of "persisting" damages itself is a non-sequitur even in a hypothetical legal system that acknowledges reincarnation. Consider a claim for "pain and suffering" inflicted in a previous life (for instance, because the defendant trampled on the victim). That claim would have to be re-evaluated because (1) most likely that pain and suffering vanished ever since, or perhaps before, that victim's prior incarnation ended; and (2) the matter would absurdly defeat the aforementioned purpose of reincarnation.