Is it true that a human can live for a while with a pig's heart but not with a chimp's heart? If it's true, then what is the explanation for this , given that the chimps are more closely related to humans both by DNA resemblance and in the phylogenetic tree ?

Their are isolated cases for both from the last century but in neither case did the human recipient live for more than a day or two. Neither are in current clinical use nor in clinical trials. I can't vouch for the veracity of all of the information in the link below but most of it tallies with what I know. … tation.htm

Leaving aside the ethics of killing chimpanzees for this purpose their hearts are not big enough to support the human circulation longer term.

In contrast a adult pig heart is the right size and more importantly their genomes can be altered to be "humanised" to stop human rejection. That said the science of xenotransplantation is still a long way off human studies and many hurdles need to be overcome including the risk of transmission of pig viruses to humans which is a real concern.